Chapter 1 Chemical shift calculations

CHAPTER 1 CHEMICAL SHIFT CALCULATIONS-f D. E. O’REJLLY Argonne National Laboratory, Argonne, Illinois CONTENTS Abstract

2

List of symbols

2

1. chemlwshl~t 1.1. Lntroduction 1.2. EJectron-Nucleus Interaction

6 6 8

2. Metho& of Calcuhztion 2.1. Perturbed Hartree-Fock Method 2.2. Variational Methods 2.3. Expansion in a Complete Set 2.4. Methods for Aromatic Molecules 2.4.1. Free+electron approximation 2.4.2. Approximate LCAO method 2.4.3. Self-consistent field method 2.5. Other Methods 2.5.1. Choice of origin at electronic centroid 2.52. &ear momentum sum rule 3. G&&t&ns 3.1, Atoms 3.2. Diatomic and Non-aromatic Polyatomic Molcculcs 3.2.1. Hydrogenmolecule 3.2.2. Perturbed Hartrcc-Fock method 3.2.3. Variational methods 3.2.4. Expansion in a complete set 3.2.5. Other methods 3.3. Aromatic Molecules 3.4. sotids 3.5. Ekctric Field E!Tects

4. l3npiricd R&s and correhtions

4.1. Additivity of Substituents Rule 4.2. Pi-eleUron Density Correlations 4.3. Electronegativity 5. summaty

16 17 :; 26 26 27 :: 31 32 33 33 :3 39 : 47 48 49 52

54 z 58 58

59 References t Based on work performed under the auspices of the U.S. Atomic Energy commission. 1

D. E.

2

O’REILLY

ABSTRACT The chemical shift of nuclei in molecules and solids is determined by the interaction of the nuclear magnetic moment with orbital currents of electrons induced by the applied magnetic field. Exactly soluble systems exist, but in most cases perturbational or variational methods must be employed. Of these, the perturbed Hartree-Foclc method yields the best results but is presently limited to small molecules, except in the case of proton shieldii due to pi-electron ring currents. The average energy denominator approximation associated with expansion of the perturbed wave function in a complete set is of particular interest because of its relative simplicity. This approximation is examined in detail and some justification of its use is given. Calculations which have been performed for atoms, diatomic and nonaromatic polyatomic molecules, aromatic molecules, and solids are summarixed. Electric field effects are of considerable importance in the chemical shifts of protons. Empirical rules and correlations are briefly discussed.

LIST

A A, 4 C, D At a am,

bHz at, h, ct, 4

av ax, bx -

BN

bm CVl

CA

E

E, En, .‘%I EN

El EZ

e#

Q

F(P,

fhd fi

G(i)

--I

OF SYMBOLS

Madelung constant variational parameters vector potential of ith electron radius of aromatic ring variational parameters variational parameters s-p hybridization coefficient hybridization parameters total field at nucleus N produced by electronic orbital motion field at nucleus N in the h direction due to orbital electronic motion orbital coefficient orbital coefficient for perturbed molecular orbital electric field strength electronic energies electric field at nucleus N first-order perturbation energy second-order perturbation energy unit vector electron afhnity first-order perturbation wave function in cylindrical coordinates coherent X-ray scattering factor trial function self-consistent field Coulomb potential operator for ith electron

CHEMICAL

8

H(i)

’

Hloc ho SO x0 *I,

x2

I IE IK f h K

K(r) K

b!Y mi ms N

NC P

&I9

ps

PW

P [PqlA Pa Pus Pz Q&)3 4s(x)

R R, R* RF ri rhV S S* k 419 2) V

CALCULATIONS

function magnetic field strength one-electron Hamiltonian of ith electron local magnetic field strength one-electron se&consistent field Hamiltonian unperturbed electronic Hamiltonian operator self-consistent field Hamiltonian magnetic perturbation Hamiltonians ionic character complete elliptic integral of second kind complete elliptic integral of first kind ionization potential current density of ith electron constant of proportionality between chemical shift and change in pi-electron density power series expansion ring current field Coulomb repulsion integral total orbital magnetic moment of electrons orbital magnetic moment of ith electron magnetic moment induced in ring s nucleus for which chemical shift is considered number of (mobile) electrons permutation operator total electronic linear momentum secular equation polynomials principal value of integral is to be taken charge-bond order matrix exchange integral p atomic orbitals secular equation polynomials radius vector between origins 0 and 0 radius vector to electroni~harge centroid from nucleus N radius vector to nuclear-charge centroid from nucleus N radius vector to nucleus ~1from origin spatial coordinate of ith electron radius vector of ith electron from nucleus N overlap integral vector area of ring s hybridization parameter trace bond orbital function electronic potential energy trial

H

SHIFT

4

e W x(P)

z Z

D.

E. O’REILLY

angle of rotation about field direction total atomic binding energy pth root of secular equation nuclear charge number of nearest-neighbour

a

4) ac

Ll Y

substituent X

YtW A

AE AEAV AEL AEt,n

perturbation denominator average energy by linear momentum sum rule energy for the ptb atomic orbital in ith molecular orbital delta function of z chemical shift of i- position proton chemical shift for proton on carbon atom ortho to X orbital energy of ith orbital Y orbital exponent Bragg scattering angle perturbation matrix element sum of products of overlap integrals bond polarity parameter bond polarity parameter wavelength/27r nuclear magnetic moment variable of integration mutual bond polarizability matrix electron density of ith electron double bond character chemical shift tensor orientation average chemical shift nuclear shielding in free atom ith principal value of chemical shift tensor XXcomponent of chemical shift tensor chemical shift operator at origin 0 diamagnetic term in aAA paramagnetic term in aAA

CHEMICAL

SHIFT

CALCULATIONS

5

paramaguetic term of aAAwith origin at electronic centre of gravity paramagnetic term of aAAwith origin at nuheus N arbitrary gauge function for vector potential flux through ring s antibonding molecular orbital ith molecular orbital first-order perturbation to ith molecular orbital electronic magnetic susceptibility of group G atomic orbital yc unperturbed electronic wave function first-order perturbation wave function first-order perturbation determinantal wave function second-order perturbation wave function angular frequency of electrons

1. CHEMICAL

SHIFT

Z. I. Introduction Soon after the initial observations of the phenomenon of nuclear magnetic resonance it was noticed that a nuclear species has si@cantly different apparent magnetogyric ratios in different chemical compounds. This effect was first reported@) for the 14N resonance in various simple nitrogen compounds, and independently@) for fsF in several inorganic fluorides. Since these initial observations, a large number of research papers have been concerned with the measurement of chemical shifts and a correspondingly large number of papers have appeared on the theory of the effect. A nucleus situated in a molecule or a solid will experience, upon application of an external magnetic field H, a magnetic field Hloc which may be defined to be the vector sum of the field H and a field H’ = - u s H produced by interaction of the applied field with the electrons of the substance:

Hloc = H - Q * H.

(1)

The quantity CJis a second-rank tensor which, by macroscopic reversibility,@) must be symmetric: a,) = uga; Q is known as the “chemical shift” or “nuclear shielding” tensor. For nuclei in molecules, u is most frequently measured for the liquid or gaseous state and the average value is obtained, where the ut are the principal values of the chemical shift tensor. a may in principle depend on the strength of the applied field or nuclear magnetic moment I_L(diierent for isotopes of the same element), but is found experimentally to be independent of both H and p. The p-independence of the chemical shift is clearly illustrated by comparison of the r4N and 15N chemical shifts of nitrogen compounds.@~ 5) For a given element, 5 values of atoms with similar bond arrangements (e.g. tetrahedral, trigonal, etc.) and bond types (e.g. u, 71,etc.) generally are grouped in a narrow interval. In the case of 14N, amines have ii values in the vicinity of zero parts per million (ppm) relative to anhydrous ammonia, cyanides (260 ppm), nitrates and nitro compounds (370 ppm), and nitrites (600 ppm). Variations of 5 within a group of similarly bonded atoms occurs due to intramolecular alterations (such as variations in charge density, bond polarity, and aromatic ring currents) and intermolecular alterations (such as differences in solvent and molecular association). Nuclear shielding in atoms exhibits a strong dependence on position in the periodic table, increasing approximately as the nuclear charge, 2. The range of shifts encountered for

CHEMICAL

SHIFT

7

CALCULATIONS

a given element increases more rapidly with Z being approximately proportional to Z3, as illustrated in Fig. l.(s) From considerations of the source of the chemical shift effect these general trends may be explained (Sections 1.2 and 2.3). A detailed interpretation of the variation of the chemical shifts for a given element generally requires reasonably accurate molecular wave functions, as well as a suitable method of evaluation of the shift. Such

INVERSE

CUBE

RADIUS

OF ATOMlC

“P ORBITAL.

(h),;

0.U.

FIG. l-Range of chemical shifts for VtiOuS &lllUltS VMSUS hVClXC cube radklS Of atomic np orbital. 0 h = 21, a (n = 31, D (n = 41, )J (n = 51, and ,A,
detailed calculations have been carried out up to the present only for relatively simple diatomic molecules (Section 3.2) for which Hartree-Fock wave functions are available. Less precise calculations have been performed for similar, as well as more complex, molecules using less reliable wave functions and more approximate computational procedures. An exceptional situation occurs for aromatic pi-electron systems, where the “ring current” contribution to the chemical shifts of protons may be evaluated with considerable accuracy from relatively uncomplicated pi-electron molecular wave functions (Sections 2.4 and 3.3). In the following section the origin of the nucleus-

D.

8

E.

O’REILLY

ekctron coupling which produces the chemical shift is reviewed. In Section 2 the various methods of calculation of the chemical shift which are available are discussed in detail. A survey of specific calculations which have been performed comprises Section 3. Calculations which have been made on solids and electric field effects are also discussed in this part of the article. In the kal section, various correlations (mostly empirical) of chemical shifts with other quantities of interest (such as nature of substituents, pielectron density, and electronegativity) are briefly reviewed.

1.2.

Electron-M&us

Interaction

The theory of the chemical shift is concerned with the magnetic interactions of nuclei with the electrons in diamagnetic molecules, i.e. molecules for which the total electron spin angular momentum is h(S) = 0. For this reason, as may be readily demonstrated,(‘) the electron spins do not contribute to the chemical shift and need not be included in the nucleus-electron Hamiltonian. Consider a molecule in an applied magnetic field H and the dipole field of the nucleus N of interest. The vector potential for the ith electron is as follows.

x rf) t where rl and rxi are the radius vectors of electron i measured from an arbitrary origin and from nucleus N, respectively, and 0 is an arbitrary scalar function of the positions of the electrons. The gauge function @ is included in equation (2) for generality. It may be used to (1) demonstrate the invariance of general expressions for the chemical shift to choice of origin (gauge) for the electrons, and (2) provide variational functions in the actual calculations of the chemical shift. The electronic Hamiltonian in the presence of the external and dipole fields is as follows: .#=

Vr + ; AC 2 + V, I

which may be rewritten in the following form:@) .@ =*o

+*1+*2,

where Jr+0= &i=-H*

-;v:+v

(da) mf-22/L. Hxrtf

mrrf , 6-h c i 3rNi -r 2imcz

2w x Sri

rNf

+

2vfQi

[2V1@ -

Vr

+

G’?Ql

WI

,

(9

CHEMICAL

SHIFT

9

CALCULATIONS

mc and rnxt are the orbital magnetic moments of the ith electron about the arbitrary origin and about the nucleus N, respectively. eti rnt = 2mci

c

rf x Vt.

i

In order to obtain a general expression for the chemical shift tensor, the change in energy of the molecule in the presence of the field I-I may be calculated from the expectation value of the Hamiltonian (3) after fvst writing ac”Y= EY in powers of the parameters p and H. Let us denote the terms in .Vl which depend linearly on p and H by @lr and HS’m, and likewise terms in S’s which are proportional to Hs and PH by Hsdps~ and &Dt’s,,a. It will be assumed that Y may be expanded in ascending powers of p and H for a fixed orientation h of the molecule with respect to the field as follows. Y = Yo +

,Yl,

+

=Qi

+

Hw!zi

+

cr%p

+

pfw3,a

+

. . .

(4)

Perturbation equations may be written by quating coefficients of equal powers of p and IY.In this way the following equations are obtained. first order in H: - .IC”li?~O = wo - Eo) YlH W first order in CL: - -.Q*o

= (al”0 -

Jo) 5,

(5b)

order pIi: =@~Yia + ~UIYQ + *oYs,~ +

~ZHYO = EO&H

+ EQHYO,

(54

E = & + H2E2a + p2Eer + /LHE~~B+ . . . and EIH = El,, = 0, since the molecule is assumed to be diamagnetic. Multiplying equation (5c)

where

on the left by ‘POand integrating results in an expression for EQH: E24

=

(yo

1 xl;

1 %r>

+


[ yl,>

+


1al”2,H

1 yo).

(6)

first two terms in equation (6) are; however, qual in magnitude and sign as may be seen by substitution of the formal solutions of equations (5a) and (5b). Ylar = - (.%?.I - Eo)-‘S’laYt) (74 The

Yl,, = -

(.@o -

EC,)-W’l,Yo

(W

into equation (6). In the following, Y~H will be designated by Yu, where I\ refers to the direction of the applied field H with respect to axes tied in the molecule. Attention will be centred on this function in most of the following discussion

D.

10

E.

O’REILLY

(see, however, Section 2.2). The second-order assumes the following form ~52,~ =

where

r~{W’o

1 W’+h

= i4V~

I *z/a

I %A>

1 Yo) -

+

energy equation Wo

2(R)

I~2g1j

(6) then

yo)}

I EN* 1 yyJ>,

(8)

BN=z2+ j

rNi

is the total field produced at nucleus N by the orbital motion of the electrons. For convenience equation (8) is written in atomic units (e = h = m = 1, a = l/c), which will be used throughout the remainder of this article. The function @ has been set equal to zero. The function !I$ is purely imaginary. This is a consequence of timereversal symmetry.@) Under the transformation t + - t, H + - H and ?P* -+ Y:. YO is real since the ground state is assumed to be nondegenerate. This requires that Yr* = - Yri, and in the following it will be assumed that Yrk is purely imaginary. This means that the first order change in electron density of the molecule due to H is zero, but the fist order change in probability-current density is fkite. The chemical shift tensor for orientation X is defined as follows: and hence

The physical significance of each of the two terms in equation (9) may be seen by calculation of the field Hi produced at nucleus N by the motion of the electrons in the applied field H. Here Hi is the field at N produced by the induced orbital magnetic moment. HA=

-

unlH=

a

(jd x

rNi),#

4i

dv,

where jt is the current density for the ith electron. ji =

I[

$_ (Y*VrY - YVrY*) - aA
(11)

The integral in equation (11) is performed over all electron co-ordinates except the ith, as indicated by the prime on dT. Atn is the vector potential associated with the external field H(= $H x rt). The vector potential term in the expression for the current density gives rise to the first (“diamagnetic”)

CHEMICAL

SHIFT

11

CALCULATIONS

term in equation (9) when substituted in equation (10). This term is due to the rigid rotation of the electrons about the arbitrary origin with an angular frequency w = a/2 H, i.e. the “diamagnetic circulation” of the electrons. The second (“paramagnetic”) term in equation (9) is produced from the linear momentum terms in the parentheses of equation (11) and which, in turn, arise from the distortion of the wave function by the applied field. This term is difficult to evaluate in comparison with the diamagnetic term which requires only a specification of Yo. As one might expect, the expression of equation (9) for the chemical shift tensor does not depend on the choice of origin for the magnetic vector potential.@* 11) A new choice of origin is equivalent to a new choice of the gauge function Q of equation (2). If the origin is changed from point 0 to point 0’ which are separated by radius vector R, then rt = ri - R and the change in the diamagnetic term u$ of equation (9) is

This change is cancelled by an equal and opposite change in the paramagnetic term in equation (9), which may be assessed by calculation of the change in ?&. Equation (7a) may be written as follows. YIA = (so - Eo)-l M,Yo,

(13)

where MA is the total orbital magnetic moment of the electrons along h. The change in YIP due to the change of’origin becomes SYl, = - ;

R x V,), Yo.

(so i

The change in the paramagnetic term of equation (9), u$,, is therefore

SU+“_

i

‘u, 1BNA(-@O-

Eo)-l( 7”

x Q),

1 Yo).

(15)

By expressing the matrix element of the product of operators in equation (15) as a product of matrix elements:
cn

1 BNA t yd
1 (F Eo

R X vt),

[ ?f’o) (16)

The

summation in equation (16) may be performed by using the relation between matrix elements of linear momentum and the corresponding coordinate. (11) (17)

12

D.

E. O’REILLY

and similarly for the ye and zt coordinates. Equation (16) then reduces to a single summation

The identity /(6*L$ #dT = +I (G*t&hi T f or a Hermitian operator L has been used to proceed to equation (18). Thus the choice of origin in the use of equation (9) is quite arbitrary and may be selected so as to simplify the calculation of O~~ This important result suggests an approach to the calculation of uAl\ wherein the gauge function 9 is chosen so that the paramagnetic term, which is relatively difficult to calculate, is made equal to zero.(rr) A sufficient condition for this to occur is that ?PrA= 0, which may easily be seen from equations (4b) and (7a) to require that

l”*+q

I

2V,@ * vg + VF@. !Po = 0.

(19)

This equation, however, is identical to equation (13) for !f’l* if the substitution YIA = @Yo is made. Hence to seek a solution of the I%& order Schroedinger equation (equation (7a)) is equivalent to searching for a choice of gauge function such that the paramagnetic term is zero. Another, but equivalent, approach is via the equation of continuity of the electronic probability density in a coordinate system rotating at the angular frequency o = (a/2)H about the direction of H.(rs) The equation of continuity(l”) in the rotating coordinate system is

The time dependence of 1 ul, 1s may be expressed by the operator M,: = wa 1ae YA 12 2iw al cl2 at = y MA 1 ul, 12.

(21)

Using equation (4) and substituting into equation (20) one obtains equation (13). Thus the equation of continuity for electron density in a magnetic field is equivalent to the Schroedinger equation for YrA, to first order in H. The chemical shift is most often measured for liquids or gases for which 3 Tr a is measured. Since the trace is independent of coordinate system the following expression for 5 is obtained from equation (9)

CHEMICAL

SHIFT

CALCULATIONS

13

The A summation in equation (22) is over three mutually orthogonal orientations of the applied field relative to the fixed molecule. The main task associated with the calculation of chemical shifts, given a reasonably accurate YO for the molecule, is the solution of equation (5a) for Yl, and then a straightforward calculation of aaA from equation (9). This may be done approximately by various procedures which are discussed in detail in the following section. Before going on to consider these various methods, however, we shall discuss several situations for which the calculation of uAa is relatively simple. Consider a cylindrically symmetric molecule at an orientation A’ with respect to the applied field such that the system has cylindrical symmetry about the field direction. In this case MrYo E 0 and hence YiAe= 0 and dA = 0. The electron density rotates as a rigid body about the field direction with frequency w = (a/2) H. The current density (equation (11)) for the ith electron is given by ji = - ; (H X 9) Pr,

(23)

where r~ is measured from an origin on the symmetry axis aligned along H, and pr is the electron density of the ith electron. Contours of constant current density and the current vector direction curves are circles around the symmetry axis.‘For an atom in a S state the expression for (I takes a particularly simple form u==;(%~~-J-~Ycl)

(24)

known as the Lamb shift.(l4) Here u is proportional to the Coulomb potential energy of the electrons at the nucleus (Section 3.1). It is of interest that if the origin of coordinates is chosen to be at a distance R from the symmetry axis, then (choosing A’ = z and R along x)

and the parama gnetic term does not vanish. The solution to equation (13) is easily seen to be Yl, = -

‘c Fi

(R x r& Yo.

(25)

It may readily be verified that for an isolated atom in a magnetic field with vector potential A that the exact solution is YA = YO exp (- i&N * r), where &? is the value of A at the nucleus of the atom. By substitution in s$, = E#A with se0 given by equation (3) this solution is verified. Such atomic orbit& are “gauge invariant atomic orbitals”. Equation (25) has

14

D. E.

O’REILLY

been generalizedcs) to the case that a solution !PIA is known for a particular choice of origin 0. If then a second choice 0’ is made ?& is related to !PIA as follows:

Equation (26) is a consequence of the requirement of translational invariance of.%‘+ = E#. An example of the use of these results is provided by the calculation of the average local field at a point N a distance R from an isolated hydrogen atom neglecting effects of electron spin.(l@ To make the paramagnetic term zero, the origin is selected at the proton. From equation (22)

(27) which may be rewritten in the convenient form

obtained by differentiating result is

the first integral with respect to E = f e-2n(1 + 2R).

R. The final (28)

The contours of constant local field or “isoscreens” are thus circles which decrease exponentially in magnitude with R.At a fixed orientation h of the radius vector R relative to the field H the local field decreases more slowly with R;at large distances it behaves as follows: my =

- a$)

(1 - 3 co@ y),

where y is the angle between H and R. Equation (29) may be derived by a power series expansion of the diamagnetic shielding in equation (9) to terms of order R-3. uyarises from the local field due to the induced orbital dipole moment of the atom which averages to zero over a sphere. A similar expansion can be performed for a group of electrons in a molecule: 071 = -d3{X& - 3(xG - RI, RJR2),

(30)

where xo is the molecular susceptibility of the electrons of the group G, and R is the radius vector from the point at which the shielding is calculated;

CHEMICAL

SHIFT

15

CALCULATIONS

the origin is located at the electronic centre of gravity of group G. The average shift becomes

Equation (31) simplifies in the case of axial symmetry of group G: (e)o = ‘&

(1 - 3 co9 y),

(32)

where AxG = ~7 - xy and y is as defined above. Another situation which is exactly soluble: an “jsolated” atom in which there is a ground state wave function with “quenched” orbital angular momentum. Consider a single electron outside of a closed shell in a pt orbital state that is quenched, i.e. the pz and pv states are at higher energies AEz and AE, respectively. With the applied magnetic field along the z direction, the paramagnetic term is zero since mzpz = 0 and the shift of the nucleus N is

(33) where the sum of the first term extends over the closed shell. With the applied field along the x direction mzp, = i(a/2)pg and equation (13) is

po -

Eo;xlz =tP#.

The solution to equation (34) is xl2 = iapy/2A&,

(34) and hence

(35) A similar result is obtained for aAA.The orientation average shift is (36) The paramagnetic term in equation (36) may be written with an “average energy denominator” AEAV (Section 2.3) equal to 2AEzAE,, (AEz + AEd’

16

D.

E.

O’REILLY

If all orbitals px, pn and pz arc doubly occupied, the paramagnetic contribution to the shift becomes zero, since the con@uration pa is a closed shell. This result neglects mixture of higher angular momentum orbitals into the ground state due to the external perturbation which produces the pstate splitting. The paramagnetic contribution to Use due to an admixed orbital will be of order h(dE*/dE,,,) times smaller than the p-state contribution where X is the square of the admixture coefficient and AE, and AE, are the pstate splitting and admixed excited state energy separation En, - Eo. In summary, a general solution of equation (13) for an electron in an atomic orbital x with quenched angular momentum may be written in the form xla = (mJAE - (ia/Z)(R x r)&y, where AE is a constant which may be determined from equation (13).

2. METHODS

OF CALCULATION

The most accurate wave functions generally available for simple molecules are self-consistent-field (SCF) functions derived from a limited set of basis atomic orbitals. Such wave functions may be expected to yield the best results for o;\* A straightforward, but tedious, approach is to solve equation (7a) utilizing a limited basis set of perturbation atomic orbitals, which are judiciously chosen to yield a minimum energy to second order. The magnetic interaction 21 produces distortions of the molecular orbital wave functions which result in the paramagnetic contribution to ul\* If equation (7a) is solved in the SCF approximation, both the direct effect of the X’r term in the Hamiltonian and the change in the self-consistent-field energy resulting from the perturbation must be included in the evaluation of !Pr* and EsAk This method of solution is considered first (Section 2.1) and then alternative procedures are discussed in which, (1) the change iu SCF energy is neglected, and (2) in addition, simple products of molecular orbitals are used for !Po (Section 2.2). Methods involving the expansion of YrA in a complete set of functions, the average energy denominator approximation and the use of gauge invariant atomic orbitals (GIAO) are of considerable importance (Section 2.3). The special case of aromatic molecules with mobile pi-electrons is calculated with the use of gauge invariant atomic orbitals or, preferably, the SCF method (Section 2.4). More approximate methods (Section 2.5) of evaluating equation (9) are useful in the estimation of overall trends and additional effects such as hydrogen bonding. These methods are primarily concerned with the evaluation of the “ditllcult” paramagnetic term in equation (9).

CHEMICAL

SHIFT

17

CALCULATIONS

2. I. Perturbed Hartree-Ebck Method In the ordinary SCF approximation(17~rs) the function YOfor 2n electrons is written as a single Slater determinant of n doubly-occupied orbit& +&

The Hamiltonian XO is as follows : (38)

where 2, is the charge of nucleus u. Equation (38) is a sum of one&ctron and two-electron operators. Variation of the orbitals 6 to yield the lowest energy requires that qcr satisfks the equation zep = &??$cr”, (3% where 20 is the SCF Hamiltonian 30 = I: (H(i) 4 G(i)) + (Eo - F CC). i

is the sum of the one-electron operators for the &h electron which occur in equation (38); c(i) is the sum of Coulomb and exchange potentials:

H(i)

where the operator Ptt interchanges the space coordinates of the electrons. The total energy EOis as follows:

i

and j

&=*~(WW+Ct)9

(41)

where the rt are the orbital energies. +t satisfies the eigenvalue equation ho(i) 4r = 9 +, where

(42)

ho(i) = H(i) + G(i).

Equation (7a) in this scheme is as follows P) (ho(i) - l Xhah = -%A

h,

(43)

where (+I,& is the first order perturbation correction to A. #IA now consists of -mA plus the change in G(i), the SCF potential, due to the first order change in the ith molecular orbital GIA(i) = -

(44)

18

D.

E.

O’REILLY

It has been assumed that +r~j is purely imaginary and, hence, the change in G arises exclusively from the exchange potential part of G(i). Equation (43) is an inhomogeneous second order differential equation, which for a molecule may only be solved approximately. Likewise equation (42) is difficult to solve. Approximate solutions are obtained by expansion of #t in a limited basis set of m atomic orbitals xp, which contain variable parameters 4s = ,-

1 X&f.

(45)

By judicious choice of atomic orbitals the basis set can be kept reasonably small and, by minimization of Eo, yield accurate orbital energies. In a similar manner the functions $I~{ may be expanded in a limited basis set of ml functions : dlAf =

*I

r,

.

x.

p-1

p

,

cpra

(46)

*

The best choice of a set of functions {xi> will generally not be identical with the set {xr). Let us consider the set {xp} to be augmented by the set {&} and expand +l,~ in terms of the unoccupied molecular orbitals 4f(i=n+l,

. . .

,(m+ml>),

which are obtained by minimizing EO with the augmented basis set (47) Placing equations (44) and (47) into equation (43), multiplying both sides by &(i) and integrating, yields

where

[m ii1 =

1CiG’)W

4; (i’) df(j’)

drf,

dTj,_

Wj’

Equation (48) consists of a set of (ml + m - n)(n) coupled inhomogeneous linear equations for the C$. Equation (48) may also be obtained with the use of time-dependent Hartree-Fock theory,(17) where the time-dependence of the Hamiltonian is provided by a rotating coordinate system as discussed in Section 1.2. The total second order energy may be calculated from equation (8):@@)

CHEMICAL

SHIFT

19

CALCULATIONS

The tist summation in equation (49) is the diamagnetic shielding and second the paramagnetic contribution. The first order energy is zero, but second order energy is changed by a change in the orbital exponents of {xi>. These atomic orbitals are most often chosen to be Slater functions the form

the the the of

times a spherical harmonic and centred on a particular nucleus of the molecule. Equation (48) may be solved for the coefficients C$ and the resulting second order energy equation (49) calculated. By variation of the orbital exponents 5 to produce a minimum in Es an optimal set {xi} is obtained and the shielding may be evaluated. However, as pointed out by Stevens, Piker, and Lipscomb, it is necessary to minimize the total second order energy including the contribution of the magnetic susceptibility EZH which, for large values of the magetic field, will be the dominant contribution. 2.2. Variational Methods Another approach to the calculation of the chemical shift is to minimize the second order energy directly with suitable variational functions rather than attempting to solve equation (7a) in the manner described above.@O) To accomplish this the quantity EZ+B is needed as a functional of the variation functions $1~ and #I~ Equation (8) is not a wholly satisfactory expression for Ezra since it is valid only if the first order perturbation equations (7a) and (7b) are satisfied. The second order energy may be computed from the difference (P 1a? 1 ?P>- (Yo 12’0 1 Yo) : 8E(p, 2T.l= P

I .@‘o+ @‘I,, + Hxui

+

pH~f2gi

-

Eo I u/>.

w

Retaining only terms of order pH, equation (50) may be written as a second order energy : WV

E2(hr,

$WY) = +

2{Wo

I+

Wl,

1so

I Ym) -

+

Eo I %)}

Wo

I .@‘uz I ‘h/J +

Wo

I *2pa

I PO>.

(51)

Equation (51) reduces to equation (8) if equations (7) are obeyed. If the first order perturbation equations (7) are satisfied for a definite choice of trial function, as in the preceding section, then one may simply minimize the total second order energy which for large applied fields will be - (Yo I MI\ I YIA) H2. If equations (7) are not satisfied then the functional

of equation (51) must be minimizd with a suitable choice of variational functions Yrr and Yla. Consider the evaluation of Es given by equation (51) in the case that YO

20

D. E. O’REILLY

is a SCF function as in equation (37). As before, each molecular orbital b will have a first order perturbation function except, in the present case, both a function iirst order in H,

and a function fist order in p, +I,,$ = ag& (51) may be written as follows:

+

Vl,(0

120

-

will be considered. Equation

Eo I Q40>}

+


I ;C”2,4a I !J’o>,

(52)

where Ym(i) or Y$(i) is the determinantal wave function found by replacing the ith orbital +t by ag& or (a/2)fi+c respectively. The first two types of terms in equation (52) may be directly evaluated using the fact that the unperturbed orbitals q$ are orthonormal, but fg,& is not necessarily orthogonal to #j (Yo

I .@lp

I hd0>

=

; {<+t I h

Ih W - j i (h1ha 1hI>>. (53)

The third term in equation (52) may be eval~ted by noting that the function (Zo - Eo) YlH(i) is equal to the sum of two determinants, one of which is Yom with the ith orbital replaced by -+Vg and the second is YtcF with the ith orbital replaced by -Vr+, * Vif. These functions arise from the action of the kinetic energy operator on Y&). After three types of integration by parts one obtains@“) (Yl,(i)

I ZO

The results of equations (53) and (54) may be combined to yield an for uA,which can be extremixed with a suitable choice of variational ggr\ and frpBy arbitrary variation of the functions of equation easily seen that 6Ea = 0 requiresthat equations (7) are satisfied. variation of Yrp and YIR yields(sO) S2EZ= 2pb

I so

-

expression functions (51) it is A second

Eo 1 w/d,

which, for arbitrary variations of the functions, may be positive or negative and thus the stationary value of uhr\ calculated by this procedure is neither a maximum nor a minimum.

CHEMICAL

SHIFT

21

CALCULATIONS

Another, less rigorous, approach is made by assuming that equations (7) are satisfkd in which case equation (51) reduces to Ea(%r) = 2
IJl”lp

I %Ir>

+

vo

and EZ is a functional of only ??‘IH.For a SCF

(55) equation (55) becomes

lJP2pa

ulo,

I yo>

Equation (56) reduces further if the molecular orbitals are of such symmetry that the quantities (& 1~N,II h) = 0 so that the exchange contribution to equation (56) becomes zero. This is the case for a diatomic molecule which has only u orbitals occupied and is equivalent to the use of a simple product of orbitals for ?I%.Variational functions which have been employed for $f are polynomials which have the following form for a diatomic molecule:(s@ fil: =

(57) with the magnetic field perpendicular to the symmetry axis (Mimction). This form of fr is suggested by the exact solution for ?P&for an atom in . equation (25) as well as, in a more general way, by consideration of the symmetry of equation (13). An example where an expansion of this kind leads to di&ulty occurs when there is an occupied pr orbital on atom A. Clearly, from equation (13), Yrz must have a node in the xy plane at atom A, which is not provided by&& (equation (57)) and the factoring assump tion (i.e. +I~{= fi& is not appropriate.tsQ $%yN

+

&‘NzN

+

b’7yN

+

&NyNzN)

2.3. Expansiim in a Complete Set Another approach to the evaluation of a solution to quation (13) may be obtained by writing equation (13) in the following form %,I = IS (so - &I)-’ I fJ’&Wn II

=

c<‘p*I E,,

0

MA I yoo) y -

Eo

I MA I ‘uo>

u*

(58)

The paramaguetic term of equation (8) then becomes@ 4 = -

2c(ul, 1MA 1yo> n

En - Eo

(59

22

D. E.

O’REILLY

Equation (59) is generally very difficult to evaluate exactly because of the appearance of matrix elements between ground and aN excited states which do not yield zero contribution because of symmetry. The so-called “average energy denominator approximation” consists of rewriting equation (59) as follows : c$ = - 2

V’o I BNAMA 1 Yo>,

(60)

~EAVA

where AEAV is an average energy which allows one to reduce the sum of the products of matrix elements to a single diagonal matrix element. For an isolated atom it may easily be shown that AEAV is always positive. The use of the approximation in equation (60) has been justified primarily by its success in the interpretation of observed shifts (Section 3.2.4). The numerator of equation (60) may be expanded using a single determinantal wave function for ?Powhich contains orbitals that are linear combinations of atomic orbitals, as in Section 2.1 (equation (45)). Then by the usual rules for evaluation of diagonal matrix elements :

-

2 (+t 1h

j#i

1h)(h 1m,a1+t>)

1

.

(61)

Equation (61) consists of direct and exchange one-electron matrix elements. It may be conveniently written in terms of the charge-bond order matrix p P/&v= F 2G

(62)

C,I

by substitution of a linear combination of atomic orbitals (equation (45)) for +t: -2 (x, I hm I x,J 8’ = LIEA~, -

‘z P,,(x, I bm I x,Xx, I m, I x,) (63) . P* Often the further approximation is made to neglect the effect of all atomic orbitals other than those centred on nucleus N.(el) Let us assume, in addition, that only s and p atomic orbitals are important in the limited basis set expansion, Let the coordinates mutually orthogonal and in cyclic order to X be designated by a and /I. In equation (63) TV,Y, p, K refer to nucleus N only and take on the values a and & and equation (63) may be seen to simplify to(21,6) -acra ‘I‘la= d*

0

c

1 $ p {Pz, + PM - P,,P&4 +

I P4

I “1.

(64)

CHEMICAL

SHIFT

23

CALCULATIONS

The following relations@r) have been used in deriving equation (64): mA x(s) = 0 mA XW

= i i x(P&

ma X&J) = -

i lj x(P,>

mA X(PA) = 0,

where x(s) is a s atomic orbital, etc. The average paramagnetic shift is

(V = -

J-j&a2($J,NPXX+ Puu + Pzzz) - 3 (PZZPUU+ PUUPU+ PzzPz2) + 3 ( I PXUI 2 + I PUZI 2 + I PZZ I 2>>. (65)

In a similar way the contribution of d orbitals to the paramagnetic part of the shift tensor may be calculated,(s) although the expressions contain considerably more terms in products of matrix elements of the charge-bond order matrix. Expressions somewhat similar to equations (64) and (65) have also been derived,@) using the valence bond “perfect pairing” wave function: (-

I)* p{‘,h(l, 2) ‘h(3, 4) . . . ‘h(b

- 1, a)),

(66)

where P permutes electrons between pairs. A typical pair wave function is #i(l, 2) = u(l, 2Nl)

P(2) - a(2) kKWd2.

~(1, 2) is the “geminal” or bond-orbital function which is written as a symmetric product of atomic orbitals. The results for o$ have been specialized for certain bonding situations, such as 1QF in ftuorobenzenes.(sl) These results will be discussed in Section 3.3. The average energy denominator approximation is, generally speaking, much better than one would expect at fist sight. AEn,. appears to be fairly constant for a given element and positive, as indicated by the data of Fig. 1. Let us examine this approximation in greater detail by considering further the LCAO-MO approximation for Ye. As in Section 2.2 we write YU = CYl,(i), where ??‘l,(i) is similar to YOwith the difference that the orbital & is replaced by +I,+ The paramagnetic contribution to the chemical shift is obtained from equation (9)

D.

24

E. O’REILLY

Equation (43) may be written as follows (ho(i) - ~6)+I~{ = mn $t.

(68)

Expanding h and 4~ in a basis set of atomic orbitals ($a = I= C,t xlc, $I~< = 2 CL{x;,J and substituting into equation (67): & = - 4 T ET; C:, Cl<
js ( Fz

I x;r>

C; C; C X,Ax,

I x;rXx,

I @an I x,>>

(6%

and equation (68) becomes E Cl&o(i) - ~0 x; = 2 C&AX,. P P

(70)

As discussed in Section 1, the solution to the free atom in a magnetic field is known and may be used to construct first order atomic orbitals x’,,~which, in turn, are used to construct +I~<. From Section 1.2 we write

* -!!!&x,(i) q&ix&9 = C’cn Aa,,

- C,r (tRP

x rs),x,(i)

... ,

(71)

where mpA is the X component of angular momentum about the nucleus of orbital 1~and Asfi is an energy denominator to be determined which may depend on both i and p. Noting that .

F V:(R, x r&, x,(i)

= “(Rr X WA x,(i) +

‘f (RF x

rt>* V2x,(i)

and substituting equation (71) into equation (70) one obtains (72) Multiplying both sides of equation (72) by mvAxr and integrating

c

Cl< rn (QxV c c

I ho - 61I m,A x,> = 2 CPr(m,~ xv I mPAx,). P

(73)

To evaluate the quantities C;,/A e(pA let us assume a minimal basis set LCAO scheme and neglect all two-centre mtegrals as before. Since all atomic orbitals for a given nucleus are orthogonal (74) where I Y’) refers to the normalized ‘orbital obtained by operating on xv with mvA and the summation is over only p orbitals on nucleus N. Of the

CHEMICAL

25

SHIFT

integrals (v’ 1ho 1p’> those with p’ # Y’are either zero or much smaller

than the $ = u’ ir~tegral(~~) and equation (74) simplifits to (75) Substituting equations (71) and (75) into equation (69), one obtains:

(xc I brrAI xc>),

(76)

where AE,,, is de&ted as follows: P A==

WC

2

c, c; c i - Y’

The approximation made previously of dropping all integrals except those between orbitals on nucleus N will now be made assuming only s and p orbitals on nucleus N. Equation (76) becomes

where Aafi are in cyclic order. In the special case that the magnetic field direction is along the radius vector between nucleus N and another m&us B in the molecule, terms of the form (XX 1bNA I x~~>(xr, 1mBA I XB> should be included in equation (77). Equation (77) is quite similar to equation (64) and becomes identical with it if all the energy denominators are equal. The energy denominator of equation (76) is a “delocalization energy” of an electron between the atomic orbital xi, and the molecular orbital & averaged over the orbitals xi. Since the maximum value of the terms inside the brackets { } of equation (77) is equal to 5 the maximum paramagnetic term for an element (neglecting d orbitals) is (4~~/3dE~~)~.Thus the constant of proportionality between AC and (l/t8>P of Fig. 1 should be 4aa/3d&, where AEAT is XI average electron delocahzation energy. Values of AEA~ ckulati from the data of Fig. 1 range between 10 and 20 eV, i.e. of the order of the first ionization potential of the molecule. Although variations in the diamagnetic term 2

26

D.

E.

O’REILLY

of equation (9) occur in various compounds of an element, except for hydrogen the largest variation in the shielding arises from the paramagnetic term. 2.4. Methods for Aromatic Molecules Aromatic molecules with one or more conjugated rings have a large diamagnetic anisotropy due to intra- and inter-ring currents of the mobile pi-electron system in a magnetic field. These non-local currents give rise to a secondary magnet& field H’, which may be calculated by equation (11) if the currents j, are known. The field H’ produces a contribution to the chemical shift, which for protons is appreciable compared to the contributions of other sources. The basic task remains to calculate the currents in the pielectron system. In all calculable theories at present the assumption is made that pi-electrons in orbitals antisymmetric to the plane of a ring can be treated separately from the remaining electrons which form an inert core. 2.4. I. Free electron approximation A simple type of wave function for the pi-electrons is an antisymmetrixed product of functions for free electrons constrained to be localized on the ring-system bond network. Let us consider the benzene molecule, approximate its shape by a circular infinite potential well, and neglect interactions between electrons. The eigenfunctions for this problem are as follows (in cylindrical coordinates)

where the centre of the ring is the circular ring. In a magnetic tion similar to equation (7a), EIH # 0. The only contribution which is

located at p = z field #iAf = 0 as noting that for to Use equation

= 0, and a is the radius of may be seen from an equa+( given by equation (78) (9) is the diamagnetic term

(79)

where IV, is the total number of mobile electrons and the line integral extends around the ring. Equation (SO) may be rewritten in a more convenient form by noting that the current in the ring from equation (11) is j = -

2

HN, cos 8 6(z) S(p - a) eg,

(81)

CHEMICAL

SHIFT

CALCULATIONS

27

where 8 is the angle between the z axis and the applied field, and e+ is a unit vector perpendicular to p in the plane of the ring (j is counterclockwise when viewed along H). The field at nucleus N is, from equation (lo),

Equation (82) is just the classical Biot-Savat expression for the field due to a current line element. The resulting local field may be expressed in terms of elliptic integrals.(~) If only the average shift is required, as is usually the case, the following result is obtained from equation (82).(24) iF(ring current) =

.Jf 67~2[(l +

NC pN)2 + $1’2

11 +

1-s

c1 _

;h;2-+

2’

;;

1

ZB 9

(83)

where IA!and 1~ are complete elliptic integrals of the first and second kind of modulus ks = 4p/[(l + p)2 4 z2]. pN and ZN in equation (83) are in units of the ring radius u and are the cylindrical coordinates of the nucleus of interest relative to the ring centre. Equation (83) gives the contribution of the ring current to the shielding of nucleus N. For (& + @/s > 1 the ring current field is closely approximated by the field of a magnetic dipole of strength ~~2s- (a/4r) * HNs cos ~9= $ a%2Nc cm 8 H located at the ring centre. The corresponding expression for 6 due to a point dipole (84) is accurate to within 5 per cent for (pi + &)r’2 > 2. Equation (84) may be alternatively derived directly from the vector potential for a point dipole. For molecules which contain more than one ring, one may compute the local field from each loop at nucleus N using equation (83) (or equation (84), at a large enough distance), if the ring currents are known. The ring currents may be estimated from the free-electron model or more detailed methods which are discussed in the following section. In another closely related approach the currents are not calculated directly, but rather the first and second order energy of the molecule in the magnetic fields of the laboratory and nucleus N is calculated. 2.4.2.

Approximate LCAO method

Soon after the proposal of the free-electron model@@ to explain the anisotropy of aromatic molecules, London@@ performed a calculation within the framework of the LCAO approximation of Htickel. An effective oneelectron Hamiltonian is expressed in a basis representation of atomic orbit&, one for each atom of the ring. Matrix elements of the effective Hamiltonian are considered to be zero, except those between nearest-neighbour orbitals

28

D.

E.

O’REILLY

(/?) and diagonal elements (ad. This approximation corresponds to a wave function which is a simple product of molecular orbitals which are a linear combination of atomic orbitals. The Hiickel model is useful for the interpretation of a number of molecular properties, although its description of excited states is inaccurate and different values of B are needed to fit different molecular properties.@‘) London’s method for the molecule in a magnetic field consists of replacing the atomic orbit& with gauge invariant atomic orbit& (Section 1.2). x,(H) = x,, exp (-

ia

A, . r).

(85)

The secular determinant then depends on the magnetic field strength and the energy levels depend on H. The secular equation has the form I~,“--

4~$vI

=09

(86)

where the energy c is in units of 8, tip, = a,, tip, = ,!I(nearest neighbours), and all overlap integrals between atomic orbitals are set equal to zero. On expansion of the determinant one obtains a polynomial in x = c - ac with coefficients of the form x&C, * * * --qi+Jcp9 which forms a closed circuit if a line is drawn for each matrix element connecting two neighbouring atoms. Only those products which contain “bonds” spaced alternately, or drawn continuously around a closed ring are nonzero.@s) The diagonal matrix elements are unchanged by the applied field. The values of the non-diagonal matrix elements may be approximated@@ as follows H/W = f exp {ia (Ale - A,) * r> ,Y,&‘ox~dr = j3 exp (i z (A/, - A,) * (R, + R,) ,

(87)

wherein r has been replaced by its value at the midpoint between the nuclei /L(r= R,,) and ~(r = RJ. Single-bond products *P,X,, are real and hence do not depend on the applied field. The closed-circuit products equal /? exp (ia@&, where n is the number of bonds in the ring s and @u= it(F) (A/, - A,) * CR/, + Rv). .

(pv) refers to an ordered pair of adjacent atoms and the sum extends around

the closed circuit. @# may be seen to be equal to the flux through ring s, since mp - R,) x (Rr + R,) is equal to the area of the triangle with vertices at the origin and at nuclei v and P. The secular determinant (equation (86)) when expanded may be written in the form(2s) p(x) = C cl&r) cos (4). *

(88)

CHEMICAL

SHIFT

CALCULATIONS

29

The sum in equation (88) extends over all closed rings of the system. In the case of a ring r which contains smaller rings, @r is the sum of the values of the flux through each smaller ring. p(x) and the qdx) are polynomials in x. Expansion of equation (88) to terms of order Hs yields P(x) = IZ Q&l n

@A,

(89)

where P(x) = p(x) - Fqdx) is the secular polynomial in the absence of the field. The roots of equation (89) to order Hz may be seen to be as follows by substitution into equation (89). x(P) = ?$) + 2 x(P) CJ,@r XP =

Q&#W”<+?h

where p is an integer ranging from one to n, the total number of atoms in the ring system, and $‘) is the pth root of P(x) = 0. The energy of the pth molecular orbital is ,$(P)= cr, + x(D$??, (91) and hence, the change in the energy of the pth molecular orbital may be written as the interaction of induced magnetic moments mr with the external field. &?I?(~) =,--+zmm, - H.. (92) * where from equations (90) and (91)t2*ssg) nb = 2&I Cx x,P(Sr Pr

* H) Sa.

(93)

$3,is the vector area of ring s; and the first sum is over all molecular orbitals, each considered to be doubly occupied. The magnetic moment of ring s may be associated with an induced current in ring s equal to j, = m&.

(94)

The ring currents circulate counter-clockwise when viewed along the component of the field perpendicular to the ring, since the induced magnetic moment is always opposite to the direction of the component of the field normal to ring s, i.e. diamagnetic. From values of the calculated ring currents, one may compute the resulting shielding using equation (10). The chemical shift due to ring currents may also be evaluated directly by calculation of the first and second order change in energy due to a perturbation matrix A with elements (P = v f 1) A cy = S&L”- b = /@e,, - +I$, + . . . ).

(95)

D. E. O’REILLY

30

From equation (87) and A given by equation (2)

(96)

e/rv = aspI

S,, is the area formed by the triangle between the origin and atoms cc and V; p is the nuclear magnetic moment and RN~ is the distance of nucleus N from atom CL.Equation (95) follows immediately upon expansion of equation (87) with the vector potential of the dipole moment of nucleus N (equation (2)) added to the vector potential of the external field. The first and second order energy in the perturbation given by equation (95) are directly calculated by the usual perturbation procedure:

(97a)

pBy is the charge-bond order matrix (equation (62)) and n(PV)CxPjis related to the “mutual-bond polarizability matrix”. JJtlrV), (._,) gives the change in bond order ppy due to ‘a change in the energy matrix element a*,,, = A,, = Re A,, + i Im AK, We,,)

=$Itrw~~.p) x

Re A,,

f

Zi%,v~t,p)

Im A,,.

(98)

As before @FLY) in equations (97) and (98) denotes an ordered pair of atoms. Equations (97a) and (97b) correspond to the diamagnetic and paramagnetic terms of the chemical shift: e(ring currents)

+

2 n~p”)~up)sp”sKp (j& +&N? + &NK+ $NP

(*p)

. 111

(99)

may be evaluated from the molecular orbitals and eigenandRrv)(xpl values of the field-free problem.(2s) Although the sums in equation (99) become very large as the number of rings increases, McWeeney(ss) has shown, by a suitable transformation, that they may be reduced to sums over one bond in each closed-ring circuit.

Ppr

CHEMICAL

2.4.3.

SHIFT

CALCULATIONS

31

Se&consistentjield method

In this method a SCF wavefunction for the mobile pi-electrons of an aromatic molecule is obtained by solution of the SCF eigenvalue equations (equation (42)). A convenient basis set of functions are ptlike orbitals centred on the atoms which are mutually orthogonal to each other at the observed molecular internuclear distances.(e’* 80) Such orbitals are obtained by solution of the “standard excited state” SCF eigenvalue equations in which each orbital is singly occupied and all the pi-electrons are in the same spin state. In an applied magnetic field, the basis orbitals xP are solutions to the standard excited state equations to order R2 if they are replaced by xcr exp (- id, - r). It is assumed that the ground state molecular orbit& may be expanded in the gauge invariant functions both in the absence and presence of the applied field. A set of fist order SCF perturbation equations is set up somewhat analogous to the procedure in Section 2.1. The SCF perturbation equations are then solved by iteration for the first order coefficients of the functions xP exp (- iaA, * r). The second order energy is then calculated (analogous to equation (49)) and the contribution of the induced ring current to the chemical shift obtained. 2.5. Other Metho& 2.5.1.

Choice of origin at electronic centroid

Concerning the arbitrary nature of the choice of origin in equation (9) the questiorr arises as to whether there exists a choice of origin for which the calculations of the paramagnetic term are especially facile. From a rigorous point of view the answer is no, but for the proton in certain types of molecules it has been show@) that the paramagnetic term is relatively small at an origin located at the electronic centre of gravity of the molecule. One would expect this to be the case if the term MA& in equation (13) is nearly equal to zero at this origin. If this is so, YIP, and hence the paramagnetic term, becomes small. Let us express the shift at the electronic centroid and designate the paramagnetic term as (uQxz. Equation (9) becomes

where & is the vector distance from nucleus N to the electronic centroid. The first term in equation (100) represents the diamagnetic circulation about nucleus N and the second is the paramagnetic term with the origin at nucleus

32

D.

E.

O’REILLY

N minus (u&)E~. One may conveniently evaluate the second term by means of the Hellman-Feynman theorem which states that the expectation values of the electric field acting on nucleus N add to zero, i.e.

(101) Hence, from equation (100)

The sum on n in equation (102) extends over all nuclei in the molecule except N. The radius vector Rs may be obtained directly from the molecular dipole moment and the radius vector Rn of the nuclear charge centroid relative to N. By definition e NJ% - W

= or.

(103)

If the contribution (I&)E~ is relatively small or zero, the paramagnetic term may be calculated from the molecular geometry and dipole moment alone. (u$)sz: cannot be expected to be small if the nucleus N has a relatively large electron density and partially occupied p- or &ike orbitals, for then MA+0 # 0 and brrAYlA will also be large. This method has been applied to proton shifts by Chari and Das(si) (Section 3.2). A method which leads to similar results has been proposed by Kurita and Ito, wherein the method of Section 2.2 is used with a variational trial function which is linear in the electronic coordinates. As is evident from equation (26), this type of variational function simply alters the choice of origin for the vector potential. Application of the variational principle in the molecular orbital approximation yields the electronic centroid of each MO as the best choice of gauge. This method has been applied by Kern and LipscombW) to a number of diatomic molecules of first-row elements. 2.5.2.

Linear momentum sum rule

The average energy denominator follows :

of equation (60) is delined to be as

Generally the ii&&e sum of equation (104) cannot be performed directly (see, however, Section 3.1). If the operator kfk is replaced by the total

CHEMICAL

SHIFT

33

CALCULATIONS

linear momentum of the electrons P, = x (Pl)r, where p is the coordinate following h in cyclic order, equation (104)‘transforms to

This expression may be evaluated by application of the linear momentum sum rule of equation (17) with the following resuWsJ X~O

fwv,

=

-

I I:

wo

(mth

@t)~4~

I Yo)

a

I I; (W+a I ‘PO) * 4

uw

Yis in cyclic order to p. The validity of this substitutional expression for AEA~ in equation (60) has been investigated for a variety of diatomic mole-

cules by Kern and Lipscomb 3.

(Section 3.2). CALCULATIONS

3.1. Atoms A number of calculations have been performed to evaluate the diamagnetic shielding in atoms given by equation (24). Atomic wave functions of varying degrees of approximation have been used: Thomas-Fermi, Hartree, HartreeFock, and various analytical forms. The calculations are summa&cd in Table 1. Experimental values of the atomic shielding are, in principle, available by two methods. The first of these utilizes the fact that the nuclearelectron (non-relativistic) interaction energy is equal to the nuclear charge TAELE1. CALJXLATIONS OF ATOMIC SHIELDINO

‘I)rpeof

systems calculated

Reference

wave function

Selected neutral

atoms and ions in rangel
izE-Fock

W. C. Dickinson, Phys. Rev. 80,563

s-term function with correlation Open- and closedconfiguration functions Hartree-Fock

E. A. Hyllcraas and S. Skarlem, Phys. Rev. 79,117 (1950) F. T. Ormand and F.. A. Matsen, J. Chcm.Phys. 30,368 (1959)

HartrebFoCk Thomas-Fermi

R

(1950)

M. L. Rust@ and P. Tiwari, J. C&m. Phys. S,2590 (1%3) A. Bonham and T. G. Strand, J. Chem. Phys. 40.3447 (1964)

34

D.

E.

O’REILLY

Z times the derivative of the total energy W with respect to 2. As a result from equation (24) one obtains(s4) matom=

---.

a2 aw

3

uw

az

From experimental values of W in an isoelectronic sequence values of oat, may be computed preferably after relativistic corrections to the total energy have been made. A second method results from the relationship between matornand the coherent x-ray scattering factor f(ps) :@s*3s) 2a2 Worn = G p

a s

f&B) dtL&

(107)

0

where P refers to the principal Vahie of the integral and PB = 2fi-1 sin 6s (A = wavelength/28, &3 = Bragg angle). Values of aatom have been calculated from experimental atomic energies by Hall and Rees(s7) and by Ellison for selected first-row neutral atoms and ions. A calculation of the integral of experimental scattering factor data has been reported for neutral bromine.@s) An investigation of the nature of the paramagnetic shielding term expressed as a sum over all excited states (Section 2.3) has been carried out for the hydrogen atom.@@ The coordinate origin for the calculation was purposely selected not to correspond to the position of the dipole for which the shielding is calculated (RN) and the position of zero-vector potential was taken at a third point ($0). Equation (9) for this situation is as follows: QZ =

;<$Oj(X-x0)(x--N)+O-

yO)@--

2(90]

bN&fO-

YN))

$0)

Eo)--~(~o)~ I cG0>,

(108)

where (m& denotes the z component of the magnetic moment operator about the position of zero-vector potential. Equation (108) gives the shielding at point N due to the electron cloud of the hydrogen atom located at the origin. The paramagnetic term of equation (108) may be written as follows:

In equation (109) the sum extends over all the excited states of the hydrogen atom including the continuum for which the sum must be replaced by an integral. The sum was carried out for the discrete excited states of the hydrogen atom. The contribution of the continuum to the sum was evaluated by difference, noting that the terms in equation (108) that contain the coordinates of the zero-vector potential position (XO, Yo, 20) must add to zero,

CHEMICAL

SHIFT

CALCULATIONS

35

since the shift must be independent of choice of gauge. The paramagnetic term of equation (109) may be written as follows : Ug=

A$($0j (x,x,g2+ YOYN &) fN I *o ) *

(110)

AE is an energy difference such that the paramagnetic term is given by equation (110). Calculations were performed for XO = 1, YO= YN = 0, and various values of xx ranging from O-00 to 4XIO. The continuum contribution to the paramagnetic term ranged from 74 per cent to 47 per cent of the total and AE varied strongly with XN. The calculations serve to illustrate the structure of the paramagnetic term when expressed by expansion in a complete set of eigenfunctions.

3.2. Diatomic and Non-aromatic Polyatomic Mole&es 3.2.1.

Hydrogen molecule

A wide variety of calculations of the proton chemical shift in the hydrogen molecule have been carried out using various procedures and wave functions. A brief summary of each of the main types of calculation is given in the following and, in Table 2, results are given for valence bond (VB) and molecular orbital (MO) types of molecular wave function. Theoretical calculations of the chemical shift may ’ be compared with the average paramagnetic term - (0~59 f 0.03) 10-s (origin at nucleus) calculated from the measured spin rotational interaction constant(40) and the average diamagnetic term (3.21 rt 0.02) x 10-s (origin at nucleus) calculated by NewelW) from a relatively accurate molecular wave function. The simplest method of calculating the. chemical shift in the hydrogen molecule occurs with the use of gauge invariant atomic orbit& (Section 1.2) in a manner similar to that used for aromatic molecules.(ls~ 4s) For the simple molecular orbital function $0 = (XI + x2)(2 + 2S)-1/2, where S = (xl 1 x2) and xp is a 1s hydrogen function centred on nucleus ~1, one obtains from equation (25) 312 = - f

(Xl -

x2) Ry(2

+

2s)~“2,

(111)

where R is the internuclear distance, the z direction is along the bond axis, and the midpoint between the nuclei has been selected as the origin. The integrals which occur in equation (9) with #lz given by equation (111) may be evaluated with the use of prolate spheroidal coordinates and zeta function expansions.(44) The final results are referenced to one of the hydrogen nuclei as origin and are listed in Table 2. Remarkably good agreement with the observed paramagnetic term is

36

D. E. O’REILLY

TABLE2 -(z-p x 10s Reference Wave function __ -O-38 Aleksandrov~16~ Simple MO8 Gauge invariant o-51 MO optimized atomic orbitals Cabaret, Didry and Guy(aa.16, exponent* Simple VBb Aleksandrovo6~ VB optimized Akksandro~r~’ exponent __ --Minimization 0.50 2.64 MO optimii exponent* of second order energy 0.53 Stephen 2.69 VB optimized I exponentC o-55 Das and Bersoht+) 2.55 MO optimized exponent* Das and Bersohx@ 2.46 VB optimized o-50 exponent* o-571 Koiker and Karpl~s(~) 2.650 BLMO* 2.67 o-59 Hoyland and Pat~(~‘, Onecentre function _-_ _Hameka(@J 2.72 Complete set 0.49 MO optimized exponents expansion 0.54 2.64 VB optimized exponent0 _‘__ 0.35 Linear momentum Kern and Lipwmb@s) MO optimized I 2.88 sum rule exponent i !q. A. Co&on, Trans. Fmrrday Sot. 33,147b (1937). bW. Heitler and F. London, 2. Physik 44,455 (1927). CS.Wang, Phys. Rev. 31,579 (1928). dS. Fraga and B. J. Ransil, J. Chem. Phys. 35,1967 (1961).

-1 1

I --

-I

obtained particularly for the MO function with optimized orbital exponent. The magnitude of the induced current calculated from equation (11) using the optimized MO function is plotted in Fig. 2. In Fig. 3 a current vector direction plot is given. At large distances from the electronic centre the electrons undergo a more or less uniform precession as in an atom. Closer to the nuclei, particularly in the region between the nuclei, the current loops are highly eccentric. Calculation of the “isoscreen” map for the optimized exponent MO has been performed by Didry and Guy(4s) as shown in Fig. 4. The map shows that the dipole approximation for d given by equation (32) is grossly in error up to 4-5 bond distances from the centre of the molecule beyond which the asymptotic formula becomes more nearly correct. Calculations of r? and 6, have been also performed for simple VB functions and are listed in Table 2.

CHEMICAL

SHIFT

DISTANCE

CALCULATIONS

FROM CENTROID.

37

0.”

FIG. i. Contours of constant probability current density in Ha calculated with gauge invariant atomic orbitals and the optimized exponent MO function. The current in atomic units is 0468 aH times the value in the figure (one atomic unit of field strength = 17.1 x KP G) and the applied magnetic field is died out of the plane of the paper.

0

0.2

FIG. 3.

0.4

0.6 lj 06

1.0 I.2 1.4 1.6 I.6 20 DISTANCE FROM CENTROID, O.U.

2.2

2.4

2.6

2.6

Vector direction map of probability current density in He.

38

D. E.

O’REILLY

FIG. 4.

Contours of constant shielding @pm) at the position of a point dipole in the vicinity of the hydrogen molecule. Calculated by variational method for optimized exponent MO. @idry and Guy.(W)

A more refined variational procedure has been carried out by Stephen using the method described in Section 2.2. From equations (7a) and (7b) for the fist order wave functions $1~ and 41~ one finds

+m&

-Y,

4 =

-

hf&Y,Z)

and +IH&, 4, z) = F(P, 4 sin 4,

(112)

where the origin is at the electronic centroid, the field is in the x direction perpendicular to the bond axis and p, 4, z are cylindrical coordinates, $e being independent of (b. Similar relations are valid for ~$1,~.By means of a power series expansion, retaining only the first two terms, one obtains 4 Hi5

=

~(azzy

+

&kzYZ)

$0,

(113)

where the factorization assumption C$IH% = ergs$0 is made. A similar expression is obtained for 41~~. UH~ may be seen to be zero from equation (i’a). By minimization of ox2 as obtained from equations (52), (53), and (54) the following expression results : 922=

-

(114)

CHEMICAL

SHIFT

39

CALCULATIONS

Results of calculation of G and (6)p by this method are given in Table 2. A similar method has been used by Das and Bersohn@s) with the exception that only the function 41Hz = i by% yz 40 (115) was used and the energy E~H

=

X+0

I*lH

1 hH>

+

($0 I*2H

I #o> +

(41~

Idl"o-

Eo I hH)

was minimized with respect to by%. These results are also listed in Table 2 for comparison. The full energy minimization procedure discussed in Section 2.3 has been performed by Kolker and Karplu~(~s) using a best limited basis set MO (BLMO) function for the hydrogen molecule. Variation functions of the form $lHz

=

iyN(a'+

bzN +

CON +

~NZN)+O

and #i,Z =

iyN[&N

+

BZN/rw

+

c +

DZN]+O

(116)

were employed and ol;n was minimized with respect to the parameters. Hoyland and Parr(47) use a one-centre wave function for Hs, which is written as a sum of products of function pairs, each proportional to a spherical harmonic. A trial function is generated by application of the angular momentum operator and the energy minimized by variation of orbital parameters. Calculations have been performed by Hameka(d*) using a complete set expansion with gauge invariant atomic orbitals for various wave functions. The inevitable estimates of average energy denominators were made. Sinha and Mukherji t4*) have solved the first order Schroedinger equation for +lH using a VB wave function. The final result is identical with that obtained by Aleksandrovo@ using gauge invariant atomic orbitals in the VB function. Kern and Lipscomb have calculated the chemical shift for the optimized exponent MO using an average energy denominator given by the linear momentum sum rule (Section 2.5) and their result is listed in Table 2.

3.2.2.

Perturbed Hartree-Fock method

A number of calculations of chemical shift have been carried out by Lipscomb and co-workers using the Hartree-Fock perturbation method. At the time of writing, calculations had been reported for LiH(ls* 50) HF,W) F2,@s) and BH.(ss) The results are in excellent agreement with experiment; generally the calculated diamagnetic and paramagnetic terms are within a few per cent of the available experimental values. In the case of a molecule such as LiH, where only o-type MO’s are occupied in the ground state, nonzero matrix elements of mA in equation (48) occur only between the occupied

40

D. E. O’REILLY

u orbitals and unoccupied *type orbitals. The calculation is begun by augmenting a chosen Hartree-Fock basis set of functions with suitably chosen r atomic orbitals. These are selected, in first approximation, to be just gauge-invariant u orbitals as in equation (25). The SCF problem (Section 2.1) is resolved with this extended basis set and the first order equations

Y

W

FIG. 5. Probability current density in LiH calculated by perturbed Hartree-Fmk method. (a) contours of COIlStaMCurrent (in a.u. 0.250 aH times the value in the figure); (b) vector direction map of current. The shaded region contains parmagnetic current loops. (Stevens and Lipscomb.( (equation (48)) are set up and solved. The second order energy (equation (49)) is then minimized by repeating this procedure for various values of the Slater orbital exponents of the ?r basis set. The current modulus and vector direction maps calculated by Stevens and Lipscomb(s@ are given in Fig. 5. As is shown in their figure, there is a region of “paramagnetic” current circulation in the vicinity of the lithium nucleus which is opposite in sense to the usual diamagnetic circulation at greater distances. The best calculated

CHEMICAL

SHIFT

CALCULATIONS

values of the chemical shifts for Li and experimental values of the paramagnetic constants. Calculations were performed to check the gauge independence of the TABLE 3.

41

H are given in Table 3 along with terms derived from spin rotational at different points on the bond axis results.

HARTRW-FOCK CALcuwnoNs FOR PERTURBED Lw

ExperiaJultal

Cdculatcd -ai)b

GPV-0

-so WG-Q

I

-;.:: 26.5 -12.8

*

I I

-18.7 -13.8

-

& 1.2 f

0.5

*See Ref. 50. % ppm.

Stevens, Pitzer, and Lipscomb also investigated the effect of neglecting the perturbation of the Hartree-Fock exchange potential due to the applied field. The SCF potential terms in equation (48) were dropped and the chemical shift calculated as before. The error introduced by this procedure amounted toasmuchas5Opercent. For molecules which have occupied w as well as u MO’s, such as HF and Fs, a more complete basis set is needed, including B and d-type atomic orbitals. Under these conditions the zeroth order energy is altered by the introduction of the new orbitals for the perturbation calculation and the entire Hartree-Fock problem must be solved. The most important contributions seem to arise from atomic orbitals which are local solutions of the first order Schroedinger equation, such as quation (71). The results are generally in good agreement with experiment, although some gauge dependence has been found because of the relatively small basis sets used in the calculations. 3.2.3.

Variational methoa’s

The most extensive calculations of this type have been carried out by Kolker and Karplus@@ for a number of diatomic molecules of tit-row elements for which BLMO functions are available. Variational functions of the form equation (116) were used for each MO. A mod&d function must be used when w orbitals are occupied. As illustrated in Section 1, when a magnetic field is applied in the x direction of a pz orbital ; 41~ is proportional to pu so as to induce current flow across the node of pt. In this case the trial function must be modified to be (y/z) +O as for the w MO in HF. Values of (6)~ for molecules which have been calculated by the minimization of the second order energy (neglecting the change in SCF potential) are

42

D. E. O'REILLY

listed in Table 4 and compared with available perturbed Hartree-Fock calculations and experimental values derived from the spin-rotational constants. In most cases the neglect of the change in SCF energy introduces an appreciable error in the calculated paramagnetic term. TABLE4. PAIUMAGNFXIC TERMOF CIDWICALSHIFT OF FIRST-ROW DIA~~VUCMOLECULES, -09 x IV

Molecule

HI Lis NB F2 LiH+C Li*H H*F HF+

Minimization of magnetic energy&

Perturbed Hartree-Fock

5.71 9.40 195-l 246.4 12.1 8.02 83.2 103.6

-

1 Experiment 5.65 4&3

805 * 2”

12.8 17.7 77.5 67.7

1;8 18.7 79.6 67.9

aBLM0 functions. bFifteen orbital basis set function. Gauge at F. Wrigin of coordinates at starred nucleus for which the result is quoted.

Values of the average proton chemical shift for the C-H fragment have been calculated by AIeksandrov,(rQ Stephen,@) and Cabaret, Didry and Guy.(M* 55) Aleksandrov performs calculations for a simple two-electron molecular orbital or valence bond wave function with gauge invariant atomic orbitals, as for hydrogen (Section 3.2.1). At fixed ionic character, Aleksandrov finds little variation (< 0.3 ppm) of the total proton shielding with carbon-atom hybridization (sp3, sp2, sp). StepherP) and Cabaret et a1.(54) use a variational function of the form of equation (113) and equation (115), respectively. Stephen employs a simple molecular orbital function of the form 90 = N{xc + xxi& (117) where xe is a sp hybridized atomic orbital of carbon and xn is a 1s hydrogen orbital with 2 = 1. At Axed polarity parameter X, ii varies little with hybridization (c O-2 ppm). At fixed hybridization, (&/dX)+r w + 7 ppm/unit of h. The origin of coordinates was selected to be at the electronic centroid and the paramagnetic term invariably was small (< 0.4 ppm) for this choice of origin (Section 2.5.1). Cabaret et a1.c54) calculate 5 for CH4, treating the contribution of each bond to 5 independently. With a bond MO of the form of equation (117) and h = 1, they obtain 3CH4) = 24.9 + 2.7 = 27.6 ppm,

CHEMICAL

SHIFT

CALCULATIONS

43

where the first contribution is that of the C-H bond of the shielded nucleus and the second term is the contribution calculated for the three neighbouring C-H bonds. This result is about 12 per cent less than the current experimental value (30.9 ppm). Stephen obtains 26.7 ppm for the C-H fragment (h = 1, @). Didry er al.@@ have calculated the average “isoscreen” surfaces for a C-H fragment and a rotating -CHs group. Such a map is shown in

Fro. 6. Contours of constant shielding (ppm) in vicinity of a two-electron C-H bond; linear combination of Slater orbitals with tetrahedral hybridization of carbon. (Didry, Guy and Cabaret.@~))

Fig. 6 for C-H. Similar maps have also been constructed for a aliphatic C-C bond fragment.@@ As with the HZ molecule, the isoscreen map for C-H shows that the dipolar approximation is quite poor out to distances of the order of 4-5 bond lengths from the centre of the molecule. Das and Ghose@r) have calculated principal components of the shielding tensor for the HsO molecule minimizing the second order energy Eax with trial wave functions of the form &t = g&, where gt is a polynomial in coordinates. Ground state wave functions used were “localized” and “nonlocalized” LCAO MO functions. Best results were obtained with a LCAO

44

D. E. O'REILLY

MO wave function calculated by Elli8on and Shull(88) (5 = 13 - 1 ppm) but the result was only 44 per cent of the experimental value (29 -7 ppm). In the evaluation of the term ($1~ 1 .%‘o - EO 1 I/&, Da8 and Ghose(8n as well as Karphrs and Kolker@@ emphasize the use of an effective electronic Hamiltonian4& such that for an approximate electronic eigenfunction $8 employed in calculations*; #O = EO$0. The evaluation of the integral (IGlliI*;--EoI

#QH)

is particularly easy with #rrfl =fi +I and ft a polynomial in electronic coordinates. In the SCF scheme (Section 2. l), the effective Hamiltonian 28 is explicitly employed. 3.2.4.

Expansion in a complete set

Calculations of chemical shifts by means of an expansion of ?ZQ in the complete set of eigenfunctions of the unperturbed Hamiltonian @8 are generally difficult to perform with known accuracy because of the infinite summation in the paramagnetic term. As discussed in Section 2.3, the average energy denominator approximation is usually used and in some cases attempt8 have been made to make the paramagnetic term small by (1) suitable choice of gauge and (2) use of gauge invariant atomic orbitals. Such calculations have been carried out by Hameka(88) for the proton shielding in the hydrogen halides HF, HCI, HBr, and HI. The ground state wave function was taken to be an antisymmetrized product of halide atomic orbitals and a hybridized bonding orbital of the form #x = (1 + 2hxSx + AS)-1’8(XH+ hXX4.x) x4x=axxnsx

+ bxxnpx,

(118)

where hx is the polarity parameter, Sx is (XH I x4x>, and XH is a hydrogen 1s orbital, xnrx and Xnpx are halide valence m and np orbitals (n = 2 for F, n = 3 for Cl, etc.), and ax and bx are the hybridization parameters, The bonding orbital dipole moment was evaluated in terms of the parameter hx with the hybridization parameters arbitrarily taken to be l/2 and l/2 43 for ax and bx, respectively. Overlap integrals were evaluated with Hartree atomic functions and other integrals by means of Slater atomic orbit& from the tables of Mulliken, Rieke, Orloff, and Orloff.(8O) Charge transfer parameters were then evaluated from the observed dipole moments. Let us denote the chemical shift operator a&O) to be as follows:

rt

’ rN6 -

(rd,drNdx 3 rNi

-

=N,@“o

-

Eo)-’ MA , (119) I

where 0 designates the origin to which rt and MA are referred. The shielding is given by the expectation value of equations (3-13) for the ground state

CHEMICAL

SHIFT

45

CALCULATIONS

wave function. This expectation value was divided into three main parts ($0 l OAA I $0)= N-YXHI 43 I xa>+ 2N-lhx

+ {~-wxx

I 4x1 I xx> + 7 C&xI do I w),

w-9

where u(H) and u(X) refer to the shift operator with the proton and halide nucleus as origin respectively, and +{x denotes a halide orbital other than the bonding XXorbital. The paramagnetic part of the Grst two terms is zero, since mlxH = 0. The first two terms and the diamagnetic part of the last terms may thus be evaluated relatively easily. Integrals were evalmted numerically using Hartree atomic functions. The only part of the paramagnetic contribution of the term with origin on the halide nucleus considered was via the excited antibonding orbital: +;x = (1 - 2&Sx + ml’s

(XH- &x4x)

Xx = (1 + AxSx)/(Ax + Sx). (121) The experimental excitation energy was used as an energy denominator. Calculated shield&s for the hydrogen halides were all within 10 per cent of the experimental values and the evaluated paramagnetic contributions range between 20 per cent and 4 per cent of the diamagnetic term decmasing in the order HF, . . . , HI. The calculation illustrates how the paramagnetic.terms can be made small by proper choice of origin. Application of equation (65) to fluorine chemical shifts of fluorobenxenes has been made by Karplus and Das.@i) Karpl~ and Das consider a C-F fragment with the bond axis as the z direction and construct bond orbitals with the fluorine pa, pn ps, and 2s atomic orbitals. The pi-bonding pz and p,, orbitals give rise to pzZ= 2 - pz Pmv = 2 -

Pus

(122)

where px and pv are the double bond characters of the n bonds. The lone pair (&) and bonding (4b) orbitals are +zP= (1 - ,)1/s x&3+ 311sx2 $b

=

((I’; 1y2 ((1- SF’2xz + sl’2 X28)+ . . . ,

(13

where s and I are the hybridization and bond ionic character, reqectively; x2 and xas are pE and 2s atomic orbitals on fluorine, and the bonding orbital contains a carbon atomic hybrid orbital not indicated explicitly. Hence pzz=2r+(I+

1x1 -s)=

1 -ts+l--ls,

since each orbital in equation (123) contains two electrons.

(1%)

D.

46

E.

O’REILLY

Karplus and Das have specialized equation (65) to the case of the fluorine nucleus bonded to an aromatic ring (py = 0) and express the change in shielding du relative to a reference aromatic fluorine compound, neglecting (1) terms quadratic in the quantities LIZand dp, (2) changes in s which are in any case small (s m O-05), and (3) the contribution of &“. Placing equations (122) and (124) into equation (65): 2a2

‘= m - 3d&

1 ((1 + Zr + sr)(Ap/2) - (1 - p7/2 - sr) AZ}. ?’ 0

(125)

The subscript r refers to the reference compound. The factor (l/r3)/AE~v was numerically evaluated from the difference in shielding between fluorobenzene and the fluorine molecule with appropriate values of Z, s, and p. Prosser and GoodmanQu) have generalized the Karplus-Das treatment of F shielding in conjugated fluorine compounds by utilizing the more general expression of equation (63) wherein orbitals on the adjacent carbon atom are included in the summations. The Prosser and Goodman result includes terms (1) for the change in the carbon-fluorine pi-bond order and (2) change in pi-charge density on carbon. Changes in sigma-bond character are not considered in this treatment and all changes in shielding in the aromatic fluorides are ascribed to changes in the pi-electron distribution. The first of the additional pi-electron terms is found from semi-empirical calculations@m to be approximately proportional to the change in fluorine pi-electron density and the second has a relatively small effect, on the fluorine shielding. Correlations of calculated pi-charge density with fluorine shielding in conjugated fluorine compounds have been observed.@s* 6%64) Karplus and Pople@s) have developed theoretical expressions for carbon- 13 chemical based on equation (63), which they simplify by neglecting all twocentre integrals. This is not quite the same approximation as that made by Karplus and Das for fluorine shieldings because orbitals centred on other atoms do contribute via the exchange term in equation (63) if only two-centre integrals are omitted. If, in addition, one assumes only s or p orbitals are used in bonding equation (63) and the approximation (XB 1 mAA

1 XB)

=

(XB

1 mBh

1 XB)

is made, equation (63) becomes

+

&a&sAB

-

pa,,,)

+

2p,,,

PgpzB}.

(126)

The sum is over the shielded atom A and all other atoms in the molecule. As before a, ,k?refer to coordinates in cyclic order to )\. Equation (126) has

CHEMICAL

SHIFT

CALCULATIONS

41

been specialixed by Karpl~ and Pople(s5) to ring carbon atoms in aromatics and by Pople(ss) to various non-aromatic compounds. In aromatics a linear dependence of the shielding on pi-electron density at carbon is derived by assuming the (l/rs)p term varies with pi-electron density due to a linear variation in the effective nuclear charge with pi-electron density. Correlations of carbon shielding with pi-electron density have been ~bserved(~~~6*). Assuming sp2 hybridization for the u bonds in planar conjugated molecules, equation (126) becomes a linear function of the free-valence index of the pi-electrons and the charge transfer parameter A for carbon-hydrogen bonds. 3.2.5.

Other methodr

The proton shielding in a variety of molecules for which the paramagnetic part of the shielding has been experimentally determined (relative to the proton as origin) has been calculated by Chan and Da@) using equation (102). The residual paramagnetic part of the shielding (&) with the origin at the electronic centroid is generally less than 10 per cent of the experimental value. A fairly reliable estimate of the proton shielding can thus be made using equation (102) with (&hrz = 0. Exceptions occur for HCN and CsHs where 4% is as large as 50 per cent of the experimental value. This is ascribed to. the effect of low-lying excited states of w symmetry, which give rise to greater distortions of these molecules in a magnetic field. The method yields poor results when applied to the shielding of nuclei with partially occupied p orbitals, such as Ns or l@Fin HF. The average energy denominator approximation with ~EA~ given by the linear momentum sum rule (equation (105)) has been applied to a variety of diatomic molecules of first-row atoms by Kern and Lipscomb@), using the BLMO wavefunction of BansUs@). In addition, the linear variational function method (Section 2.5.1) of Kurita and Ito was investigated for these molecules. Neither method is at all accurate in the case of a heavier nucleus bonded to a lighter nucleus. In the case of a lighter nucleus (particularly hydrogen) bonded to a heavier nucleus, both methods give results of the correct sign and are within a range of 30 per cent of each other and the experimental value for the cases studied. Stevens, Kern, and Lipscomb have investigated the proton shielding in HCo(CO)r using the above method which, in the limit of large atomic number for the atom to which H is bonded, reduces to (127) which is obtained directly from equation (100) by assuming ((I$&~ = 0 and the electronic centroid to be at the Co nucleus. The large shielding of the

48

D.

E. O’REILLY

proton in HCo(C0)4 (45 ppm) can be accounted for with a Co-H bond distance that is close to the sum of covalent radii rather than the anomalously small values suggested by earlier workers. 3.3. Aromatic iUolecules The initial calculation of the ring-current shielding (Section 2.4) of protons in benzene was made by Pople.(“) This shielding was estimated with the free-electron model and the point-dipole approximation for which (equation (84))

(129) Placing u(ring radius) = 2.65 and R(proton distance) = 4.73 one obtains 1= -1.75 ppm, which is in fair agreement with the observed shift between benzene and ethylene (- 1.5 ppm). More extensive calculations(‘s) were performed on alternate aromatic hydrocarbons assuming that the effect of each closed six-membered ring could be represented by a benzene-point dipole in a magnetic field; rough agreement with observed shifts was obtained. The LCAO theory developed by McWeeney (Section 2.4.2) has been applied to benzene and naphthalene. For benzene, McWeeney@) finds that the ring-current shift (equation (99)) is given by a=

- 2#la2$

K(r) (130)

K(r) = $ (1 + g2 + g4

+ ...

1

with r in units of a. At the position of the proton the shielding given by equation (130) is 80 per cent higher than that given by retaining the first term alone in K(r) (point-dipole approximation). For naphthalene the shielding is a sum of two terms similar to that in equation (130), but the point-dipole strengths are 1.093 times the benzene value. The quantum mechanical LCAO result of equation (130) is better represented by the field of a finite current loop (equation (83)) than that of a point dipole at short distances from the ring centre. The calculation of ring currents in the SCF approximation (Section 2.4.3) has been developed by Hall and Hardisson@@ and applied to benzene, naphthalene, azulene, and several heterocylic aromatic compounds.@) The shieldings calculated by the SCF method do not differ greatly from those calculated by the LCAO method for naphthalene, and in azulene differences only of the order of 5 per cent occur.(sO) However, for molecules such as 1-methyl-Zpyridone (I) large differences calculated by the two methods do

CHEMICAL

SHIFT

CALCULATIONS

49

occur (20-70 per cent); the SCF method generally yields smaller ring-current shifts than the LCAO method.

Caralp and Hoarau(‘4) have calculated the proton shielding due to the ring-current in benzene, using the more complete LCAO method of Goeppert-Mayer and Sklar. The one- and two-electron energy integrals are field dependent with gauge-invariant atomic orbitals; the London approximation (equation (87)) was made and the dependence of the energy on @ evaluated. Semi-empirical values for the molecular integrals yield a value of 32.62 ppm) that is comparable to the McWeeney method (2.50 ppm)@@ or the free-electron model (2.7 ppm).(75) Waugh and Fessenden (7s)have calculated the ring-current proton shielding of methylene groups in 1,4-decamethylenebenzene (II) and 1,2-hexamethylenebenzene (III). The shielding for the methylene groups more than two carbon

atoms removed from the aromatic ring should be essentially the same as in a saturated hydrogen plus the shielding due to the ring-current field. In II the ring-current shielding for such methylene groups is positive, but in III it is slightly negative. The calculated shieldings are in rough agreement with the experimental results.(7s) For the protons situated in methylene groups attached to, or one group removed from, the benzene ring the “inductive” effect of the benzene ring on the proton shielding must be taken into account (Section 4.1). 3 .a. soll?J!s Only a relatively small number of measurements have been performed of chemical shifts in the solid state. The chemical shift in solids is often masked by splittings and line broadening produced by nuclear dipolar and quadrupolar

50

D.

E.

O’REILLY

interactions. Measurement of the chemical shift of nuclei situated in a non-cubic environment with only partial motional averaging of the chemical shift as in a solid or liquid crystal will yield information on the anisotropy of the shielding tensor. Calculations of the anisotropy of the W chemical shift in the CO:- ion have been carried out and compared with experiment by Pople.(rs) Extensive chemical shift measurements of alkali and halide nuclei in the alkali halides, including pressure dependences, have stimulated calculations of the chemical shift in these materials. There have been proposed essentially two basic theories of the chemical shift in the alkali halides. Both of these use the average energy denominator approximation and valence bond wave functions. Yosida and Moriya(W consider, in zeroth approximation, the crystal to consist of ions with no covalent bonding. The cubic crystal field will deform the closed-shell wave function of the halide or alkali ion, but this effect is neglected. The electrostatic interaction between the ions will result in some charge transfer between the alkali and halide ions that will give rise to a paramagnetic term in the chemical shift. Yosida and Moriya describe the charge transfer by mixing into the ground-state function, excited states Ii/n in which a halide electron with azimuthal quantum number ml has been transferred to a metal orbital. Such mixing gives non-zero matrix elements of bNh and rnA between the ground state and excited states &, of the crystal, in which a closed shell electron of the halide ion with rn; = ml f 1 has been transferred to the metal ion. Straightforward evaluation of the matrix elements in the numeration of equation (59) yields, for the halogen nucleus (X) ox=

2 xsz 1 +-3 AEA, 0 7; x

(131)

and, for the metal nucleus, ~M=----p 2 x2,4 -31 3 AEAV 0 rN

M’

(132)

where X is the mixing coefficient, z is the number of nearest neighbours, up is the s-p hybridization coefficient of the metal orbital, and subscripts X and M refer to halide and metal, respectively. Yosida and Moriya approximate the energy denominator by the difference in electrostatic energy upon electron transfer (2A - 1)/R, the electron affinity 8 of the halide and the ionization potential 4 of the metal atom AEA, w (2A - 1)/R + 8 - 4,

(133)

where A is the Madelung constant and R the interionic distance. Although the AEA, denominator can be estimated in this way, there is a basic difficulty in the comparison of equations (131) and (132) with experiment so as to evaluate

CHEMICAL

SHIFT

CALCULATIONS

51

the parameter A; namely, the shielding of the free gaseous closed-shell ion is not known. Values of the mixing parameter X obtained, assuming no paramagnetic shielding for the aqueous ions, range from 1 to 5 per cent. An explanation of chemical shifts of lead compounds has been proposed by OrgeU7s) on the basis of s-p mixing in the Pbs+ free-ion ground state . . . (5@0(6s)a due to a nonsymmetric environment in the solid state. Kondo and Yamashita(79) consider only the ground-state wave function of the crystal as a single Slater determinant of N orthogonalized atomic spin orbitals PO = &~(-lW~&(/J)

(134) P

The +p are obtained from the atomic orbitals of the ions by the following transformation : & = 2 (1 + qt”x, (135) times a spin wave function a or p, where S is the overlap matrix with elements s = (x, I XJ - $W. The paramagnetic contribution to the chemical shift gy be written as follows (equation (61)):

(136)_ Equation (136) has been expressed in the atomic orbitals xv by Ikenberry and DaW) by use of equation (135) with the following approximations : (1) only the outermost occupied s and p orbit& of the ions are included, (2) only overlaps on nearest neighbours to the nucleus of interest are considered, (3) the expansion of (1 + S)-lj2 is made only up to second order in the overlap integrals, and (4) only matrix elements of l/r: involving at least one xv on the metal nucleus are retained. Equation (136) may then be expressed as follows : 16as 1 A 09 aa=(137) AEAV

0&

'

where A is a sum of products of two overlap integrals between the metal and halide (I, s orbitals and s halide orbitals. Small additional terms occur which involve matrix elements of l/r; between halide and metal orbitals, which are of the order of 1 per cent of the total. Overlap integrals and the matrix elements of l/r; were evaluated using Hartree-Fock halide and metal free-atomic orbitals. The average energy denominator for Rb+ and Br- in equation (137) was evaluated by calculation of u& at three values of R and the Aa versus pressure data of Baron with the assumption that AEnv is

52

D.

E.

O’REILLY

independent of R. This procedure yields d&(Rb+) = 12.0 eV and d&@-) = 5 - 2 eV. In a later paper@Q Ikenberry and Das obtain a nearly constant Lt&(Rb+) for RbCl, RbBr, and RbI indicating that the assumption di& independent of R for a given ion is valid. By comparison of the experimental do data (relative to aqueous ion) and the calculated up, the shift of the aqueous ions relative to the free gaseous ions are estimated to be +194 ppm for Br and +63 ppm for Rbf. The estimated absolute shifts of the aqueous ions for Rb+ are found to be nearly independent of halide and agrees well with a calculated value taking into account overlap with the oxygen lone-pair orbital of the Hz0 molecule in an octahedral model for the Rb(HsO)+ complex ion. Electron orbital contributions to the Knight shift in transition metals, which arise primarily from the paramagnetic term of equation (9), have been described.(srJp*4) In the simpler metals, the Knight shift is produced by electron-spin polarization as mentioned in Section 1.2, but in the transition metals important orbital contributions occur. 3.5. Electric Field Efects There has been considerable interest in the effect of an electric field on the chemical shift, particularly for protons. Electric fields produced by polar groups or localized charges can be estimated from suitable models. The initial calculation of the effect of electric fields was performed by Marshall and Pople@s) for the hydrogen atom. In the presence of static magnetic and electric fields the wavefunction may be expanded in powers of H and E and perturbation equations for the coefficients may be solved.@s) By means of equation (10) the chemical shift in powers of E may be determined. Choosing the electric-field to be in the z direction, the shielding is as follows:

(138) u-~~=~,,=;[I--~E~]. The decrease in shift with E2 is due (1) to an expansion of the charge density about the proton resulting in a decrease in the diamagnetic shielding and (2) to a distortion of the wave function proportional to zE which gives a “paramagnetic” contribution to uzx. For systems that do not possess an inversion centre of symmetry such as the HX molecule, there will be a linear term in the electric field strength dependence of the shielding. The magnitude of this term has been estimated

CHEMICAL

SHIFT

CALCULATIONS

53

by Buckingham(*~ from equation (138), considering the E field to arise from a uniform external field plus the field of a point charge q located at a distance R from the proton. The average shift then becomes a2 b=----

3

8814

E *‘I 108 R2 ’ - 216

E2,

(139)

where Ez is the component of E along the bond axis. The quantity q/R2 was determined empirically from (1) solvent effects on the chemical shifts of protons in polar molecules and (2) intramolecular electric field effects on proton shifts. A perturbation calculation of the linear electric field effect has been given by Fraenkel et uZ.@s)and by Musher@@ for a C-H bond, The mixing of the ground-state MO + = (2 + 2W’YXV + xr) with the anti-bonding excited-state MO +* = (2 - 2S)-1/2(x 0 - xn) by the electric field is considered. xa is a u carbon s-p hybrid orbital and xn is. a hydrogen 1s orbital. Fraenkel et al. spec&ally consider the case of a charge q localized in the carbon w orbital, while Musher considers the effect of a uniform external field. In both calculations the mixing coefficient is determined to first order in the perturbing field. The change in charge density on hydrogen due to the admixture is calculated, and the change in the diamagnetic shielding due to the change in charge density is evaluated. Paramagnetic contributions due to the bond polarization are neglected. Fraenkel et al. obtain

where KA,, and &h are Coulomb repulsion integrals and AE is the energy separation between the bonding and antibonding states. Equation (140) predicts a linear dependence of proton shielding on carbon pi-electron charge density. Such a charge dependence of proton shielding has been observed (Section 4.2) for a number of aromatic ions.@*, So-@%The reason for the relatively large linear electric field effect is due to the strong dependence of the proton shielding on the bond-polarity parameter h (Section 3.2.3), which is appreciably altered by a strong electric field along the bond axis. The linear electric field effect gives sign&ant contributions, both inter- and intra-molecular, to the proton shielding of molecules containing polar groups.’

54

D.

4. EMPIRICAL

E. O’REILLY

RULES

AND

CORRELATIONS

4. I. Additivity of Substftuents Rule Simple additivity rules have been noted for proton chemical shifts. The proton chemical shifts of disubstituted benzenes may be expressed as the sum of the shifts for the monosubstituted molecules.@419%9s)Fora 1,klisubstituted

benzene (IV) the chemical shift of the 2-position proton benzene) may be expressed simply as 82 =

&(X)

+

&n(y),

8s (relative to (141)

where 60(X) and S,(Y) are the shifts of the ortho- and me&positions of the monosubstituted benzenes. Deviations from additivity amounting to fO* 1 ppm occur, this amounts to about 5 per cent of the total shift. A modified equation (141) which includes a “polarizability parameter” y(X) is as follows 82 = 60(X) + r(X)&m _ (142) Equation (142) is found to represent the shift data to within &0:02 ppm. The empirical quantity r(X) is near unity and is a function of the substituent X. The substituent effect in monosubstituted benzenes on the ring-proton shifts is an order of magnitude larger than it is for protons separated by an equal number of carbon atoms in saturated hydrocarbons. The effect is associated with variations in the pi-electron system produced by the substituent. For a monosubstituted benzene at the para-position proton, W and 1gF chemical shifts are nearly proportionaP7@) and the shift at this position arises primarily from the change in pi-electron density produced by the substituent (Section 3.5). Meta-position proton shifts are relatively small and are probably influenced by pi-electron densities on neighbouring-position carbon atoms. Ortho-proton shifts are relatively large and tend to be proportional to the para-proton shifts with notable exceptions. In disubstituted benzenes the additivity rule is not so well obeyed for protons on positions adjacent to substituents which are ortho to each other. The parameter r(X) is essentially a polarizability factor relating the change in shift of a proton ortho to X due to the perturbation of a substituent Y para to X. In addition, r(X) is proportional to &(X).@s) If the shifts produced are proportional to the local pi-electron charge density, then as pi-electron charge density is added to the position ortho to X it becomes less susceptible

CHEMICAL

SHIFT

CALCULATIONS

55

to perturbation by Y. This effect is attributed to the combined action of inductive and mesomeric effects of the substituents.(Os) An additivity rule similar to equation (142) has been observed to hold for y-substituted pyridines (V).(g@)The a-proton shift is proportional to the Y

shift of the proton meta to Y in substituted chlorobenzenes. a- and B-proton shifts of nine rsubstituted pyridines were calculated from equation (142) with mean deviations of f0 ~06 ppm and fO- 11 ppm, respectively. Additivity effects have also been observed in substituted ferrocenes.(lO~ Additivity rules are also known for an acyclic system of the type Y-CHPX and less reliably for the type XYZ CH. The chemical shifts of the methylene protons in twenty CH&Y molecules relative to methane have been reproduced to within &O-O5 ppm by an equation of the form S=&X)$_W), (143) where S(X) and S(Y) are constants characteristic of the substituents X and Y, and are derived from mono- and d&substituted methanes.oOr) More extensive additivity relationships hold for special systems such as phenyl-substituted methanes +SCH4-c and ethanes &CHS-~CH~ for which each added phenyl group (4) produces a nearly constant increment (1.45 ppm in the a and B carbon for 4zCH4-z, 0.45 ppm for CHs in &CHs-J&s) atom proton shift, respectively. This result is illustrated in Fig. 7 for the sequence of 46compounds &CHa&Hs (x = 0, 1,2,3).(la)

NUMBER

FIG. 7. Chemical

OF PHENYL

RINGS

shift of methyl group protons of phcnyl substituted ethanes MHwCHs versusx at 40 MC/S.

56

D. E. O’REILLY

Aside from basic theoretical interest, additivity rules allow empirical calculation of the chemical shifts in model compounds with a certain degree of reliability. Two possible structures for the cyclic dimer of I,ldiphenylethylene are 1,1,3-triphenyl-3-methyl indan (VI) and 1,1,2-triphenyl-2methyl indan (VII)

Structures VI and VII may be distinguished on the basis of the shift expected for the methylene protons of the five-membered ring. The shifts of the a and fi carbon atom protons of indan (VIII) (- 1.72 ppm and -0.80 ppm

respectively relative to cyclohexane) are perturbed by (1) the inductive effect of the phenyl substituents and (2) the ring currents of these substituents. The relative magnitudes of these contributions are given in Table 5, where the ring current shifts were calculated using equation (84). Structure VI is in considerably better agreement with the observed shift in harmony with chemical evidence that VI is the correct structure. Shifts listed in Table 5 are average values for the two methylene protons. The calculated difference in shift of 0.15 ppm (due to the ring current contribution) agrees well with the observed value of 0.22 ppm.(rs*) TABLE5. CALCULATED METHYLENB PROTONSHIFTSOF STRIJ~ ANDVIP

VI

Calculated Structure

Inductive

VI

-0.30

1 Ring current ( i

-0.10 1 I aRelative to cyclohexane in ppm. VII

Total

Observed

57

CHEMICAL SHIFT CALCULATIONS

4.2. Pi-electron Density Correlatips An empirical equation for aromatic molecules relating the change of pielectron density on the ith carbon atom dp and chemical shift of the attached proton has been proposed as follows: & = KApr,

(14)

where the constant of proportionality has an average value of about 8 ppm/ unit of charge. Equation (144) has been observed to hold for lrJC shifts as well for a series of relatively simple hydrocarbon ion@@ as illustrated in Fig. 8. 50-. 40E B . z E 6.7 = .I! E c’ ” ”

”

0

30zo10 o-IO

-

-20

-

-30

-

-0

Z .u

I

I 0.0

0.9

1.0

I 1.1.

I I.2

FIG. 8. Carbon-13 and proton chemical shifts of the series C,Ht, C,H,, C,H;. and CBH2,- relative to benzene versus pizlectron density on carbon calculated from pi-electron to carbon atom ratio. (Spies&e and Schneider.(l~~)

Generally, the charge densities are not known but may be calculated by various molecular orbital methods. Correlations between calculated charge densities and observed chemical shifts have been realized for monofluorinated aromatic molecules (rsF shifts),(r04) 19F, W, and rH shifts in substituted benzenes(rmJ~) and proton shifts of five and six-membered heterocyclic molecules.( 107)Such effects are predicted by equation (140) for protons, equation (125) for IsF, and equation (126) for r*C. A related e&t is observed in the correlation of r@Fshifts in para-substituted triphenylmethyl cations with molecular resonance stabilization energies.(rOs) 3

D.

58

E. O’REILLY

4.3. Electronegativity The difference in chemical shifts between methyl and methylene-group protons in substituted ethanes CHaCHaX has been observed to correlate with the electronegativity of the substituent for the halogens. Shoolery and Daileyosg) proposed a linear equation relating the electronegativity of the substituent X and the quantity S(CHs)--G(CHa). In an extensive study of proton shifts in haloalkanes, Bothner-By and Naar-Colin(llo) found a linear correlation between the molecular dipole moment and change in a-proton shielding between the haloalkane and alkane molecules. These effects may be ascribed to changes in the C-H bond polarity parameter by the substituent, both by inductive and electric field effects. As noted by Bothner-By and Naar-Cohn, however, the a-proton shielding increases with increase in electronegativity in the cyclohexyl halides. This effect is ascribed to the combined effect of resonance structures such as IX and X (which might be expected to increase importance with halide atomic number and size of R, R’, and R”) H H+ H+ H I

I

R-C=C

X-

R-C-C=X-

I I

R’ R”

6

(IX)

as well as an increased contribution to the shielding from bond susceptibility anisotropy (equation (32)) in the cyclohexyl halides. A linear correlation between the electronegativity of X and the W shielding of the carbon bonded to X in monosubstituted benzenes has also been noted.@‘) An empirical correction for the bond anisotropy effect improved the correlation. Explanation for such correlations may be found in the expected proportionality of 13C shifts to bond polarity parameter (Section 3.2.3). The bond anisotropy effect appears to be operative in determining shielding, but is often difficult to evaluate quantitatively due to an inadequacy of equation (32) within several bond distances of the anisotropic group (Section 3.2), as well as a lack of reliable data for the anisotropy of bond susceptibilities.(lll) 5. SUMMARY

A variety of methods for the calculation of chemical shifts in molecules have been developed. Of these the perturbed Hartree-Fock method is the most accurate, but at present this method is limited to small molecules for

CHEMICAL

SHIFT

CALCULATIONS

59

which the SCF method is practical. Other, more approximate, variational methods applied to proton shieldings have demonstrated the importance of quantities, such as the sigma-bond polarity parameter and pi-electron density on the adjacent atom in determining the magnitude of the shielding. Indications of the role of parameters such as pi-electron density, free-valence and bond polarity for nuclei other than hydrogen are obtained with the average energy denominator approximation. However, the elucidation of factors determining the shield@, particularly for nuclei other than hydrogen, appears to be far from complete.

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