Linnett's double quartet theory and localised orbitals

Journal of Molecular Structure (Theochem), 152 (1987) 319-330 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

LINNETT’S

DOUBLE QUARTET

THEORY

AND LOCALISED

ORBITALS

BRIAN J. DUKE Faculty of Science, University College of the Northern Territory, GPO Box 1341, Darwin NT 5794 (Australia) (Received 6 February 1987)

ABSTRACT The correspondence between the Linnett double quartet (LDQ) theory and the centroids of charge of localised orbitals is extended to utilise Linnett’s concept of formal charges, to discuss distortions from purely tetrahedral arrangements, to use restricted as well as unrestricted Hartree-Fock wave functions and, finally, to discuss a common class of closed shell molecules. The implications of this apprcach for the understanding of the computational non-paired spatial orbital (NPSO) method are explored.

Linnett’s double quartet theory (LDQ) [l, 21 has made some valuable contributions to the teaching of chemistry and to the visualisation of electronic structures. Unfortunately the connection of LDQ with standard quantum chemical methods has sometimes been missed and indeed some authors imply that this theory is in some sense non-quantum mechanical [3,4]. This is in spite of several calculations using the non-paired spatial orbital (NF’SO) method - the name given to LDQ in its computational aspect. These show that for systems such as ally1 [ 5, 61, benzene [‘I, 81, diborane [ 91, ozone [lo] , etc., the method can achieve energies lower than the molecular orbital method. The connection with molecular orbital theory for open shell systems such as nitric oxide and oxygen is clear from Linnett’s [2] original paper. This was put on a firmer basis by Hirst and Linington [ll] who transformed the molecular orbitals of these two systems into localised orbitals. The positions of maximum electron density of the localised orbitals closely fit the “dots” and “crosses” of the Linnett structures. Leroy and co-workers [12-M] have extended this approach for open shell systems. There is a similar correspondence for closed shell molecules with a single Lewis structure. Here the molecular orbitals are transformed into localised orbit& using the Foster and Boys [19] method, which maximises the distances between the centroids of charge of the orbitals. These centroids are then identified with the Lewis “dot-cross” pair. For open shell systems the unrestricted Hartree-Fock (UHF) 1201 molecular orbital method is used. The two sets of spin orbitals are localised separately. The centroids of the two sets are identified with the “dots” and “crosses”, respectively of the Linnett structures. As Leroy [16] says: 0166-1280/87/$03.50

o 1987 Elsevier Science Publishers B.V.

320

“Thus, it can be stated that the Boys localisation procedure is the mathematification of the qualitative Linnett’s theory. Therefore we find the exact meaning of the symbols used by Linnett (x and o) for representing the electronic structure of molecules. We may assume that they represent not localised electrons but centroids of charge of localised orbitals.” A good summary of the work of Leroy and co-workers can be found in the text by Daudel et al. [21]. The essential idea of the LDQ theory is that the Lewis octet is replaced by two quartets, one of each spin. The four electrons of the quartet are separated by charge and spin correlation. The two quartets are separated, where possible, by charge correlation only but will come together in structures like methane to concentrate electrons in the bond regions. In many cases the two quartets need not be superimposed and LDQ theory gives a distinctly different structure to the Lewis octet theory. It is convenient to divide LDQ structures into four classes: (A) Structures where LDQ is identical to Lewis octet theory, such as CH4, F?, HzO, etc.; (B) Open shell structures such as NO and O2 where a single LDQ structure can be written, e.g. (a) NO has 6a and 50 valence electrons. The 6a are distributed as two tetrahedra sharing an edge

’

X

Nx

X

Ox

X

The 50 are distributed

as two tetrahedra

sharing a face

0 0

N

Combining

0

0

0

these gives 0

X 0

X

N

x

0

X

0

X

0 X

0

which was abbreviated

by Linnett

0

X

N

as

X

0

A line here represents a pair of electrons of opposite spin but they are not in the same region of space. If the pair is concentrated at a point a heavy line is used. (b) O2 with 6a and 6p valence electrons gives the Lewis structure

321

but this corresponds to an excited state. The open shell structure of 7arand 50 valence electrons is lower in energy. 7a electrons are distributed as two tetrahedra sharing a vertex X

X X

0

x

X

0

X

X

giving the LDQ structure 0

X ox

0

X

X

x0

0

0

X

X

x0

O

:

X

X X

0

(C) Closed shell structures which in valence bond. terms have to be described by two resonance structures, usually symmetry equivalent, but which can be described by a single LDQ structure. Good examples are benzene and ozone. Resonance

LDQ

In these cases electrons of (Yspin are distributed as the pairs of one canonical structure and electrons of /I spin are distributed as the pairs of the other canonical structure. (D) All other LDQ structures where resonance between LDQ structures is still required. Class (A) will not be considered further since LDQ and Lewis’ theory agree. Class (D) appears to be a vague class, but since the need for resonance is not removed, LDQ structures are now rarely used for molecules in this class. Class (D) will not be discussed further. All important LDQ applications fall into class. (B) or (C). Leroy and co-workers discuss class (B) only. The

322

purpose of this work is to look at class (B) molecules in a slightly different way and to extend the localized orbital approach to class (C) molecules. OPEN SHELL SYSTEMS

Introduction The UHF wave function is given by * =

co: c @? c

- *- *G:, QSQP & P ***GE,PI

(1)

There are n, and na electrons of 4 and 0 spin, respectively. The spatial orbital sets are different. Each set can be subjected to a unitary transformation, leaving \kinvariant but localising the orbitals. This gives * =

[eyce;c

. . . . eiac

efpegp...e$p]

(2)

where ea = La #*

(3)

eP = LP @

(4)

The method of Foster and Boys is just one such unitary transformation but it is computationally simple, requiring only one-electron integrals, and it gives centroids of charge which have the correspondence to the LDQ pictures. All the work of Leroy and co-workers appears to use this method. The RHF method An alternative method is the restricted Hartree-Fock method (RHF) for open shell systems. Here as many electrons as possible are paired. The RHF wave function is given by * =

[~1(Y~1P~2(Y~2P...~“p(YdnpP~nS+1

Q...Q”aQ]

(5)

assuming n, > np 2n, electrons are paired. As with UHF the orbitals can be divided into two sets occupied by (Y and 0 electrons, but now the sets have the subset $J~. . . qb,, in common. Again we can subject the two sets to unitary transformations to give a wave function formally identical to eqn. (2) but with the local&d orbitals 0: and ef slightly different from the UHF case. A number of systems have been studied by both UHF and RHF. The basis set is the MIDI-3 basis of Huzinaga and co-workers [22, 231, augmented with d polarisation functions. This is a ls(4GTO), 2s(2GTO), 2s’(lGTO), 2p(2GTO), 2p’(lGTO), 3d(lGTO) basis. It is therefore a split valence plus polarisation function basis. Leroy and co-workers use minimum basis and ST0 4/31G. The latter is similar to the basis used here but without polarisation functions. In Table 1, as an example, results are presented for 02. The methods give very similar results. This, while not surprising, taken

323 TABLE 1 UHF and RHF resultsfor oxygen (R = 121 pm) 0

X ;T&x

0

x

O-O r,

/

0

0

0

X

X

r (pm) 0 (“)

or,

0

UHF 31 105.2

RHF 31 105.2

r, (pm) rl (pm) e (“)

UHJ? 65 27 158.7

RHF 65 25 159.7

with the results using different basis sets, supports the view that the correspondence of charge centroids with Linnett’s “dots” and “crosses”, is not an artifact of the method of calculation. The distortion of the quartets from tetrahedral angles will be discussed later. Symmetry

equivalenced

RHF method

For systems that do not have a half-filled open shell both UHF and RHF are inappropriate. For nitric oxide, for example, the configurations

and

are equivalent. The internuclear axis is taken as z. A two-determinant wave function is required. The symmetry equivalenced restricted Hartree-Fock (SERHF) method discussed by Guest and Saunders [24] can be used for this case. The transformation to localised orbitals is then carried out on one of the determinants. UHF can be used but the degeneracy of the orbitals in nitric oxide for example is lost. Centroids of charge by the two methods agree closely for a number of molecules. The method used by Hirst and Linington [ll] for nitric oxide was this twodeterminant method. The SERHF method is used for all results reported here for molecules with other than a half-filled open shell. NPSO wave functions Before presenting other results by this approach, it is interesting to analyse the effect of this approach on the NPSO computational scheme. A NPSO wave function is constructed from “bond orbitals”, built from atomic orbitals on two centres, and from “single centre orbitals”, built from atomic orbitals on one centre. These orbitals would have centroids of charge close to those of the localised orbitals discussed here. “Bond orbitals” and

324

“single centre orbitals” would arise from truncation and modification of the UHF localised orbitals. This is equivalent to a variation of the UHF delocalised orbitals. Such a variation is allowed but moves the wave function away from the optimum. Thus we can conclude that (6)

ENPSO a J&IF

For open shell systems in class (B), unlike closed shell systems in class (C), a NPSO calculation is inferior to the best MO calculation. Since no NPSO calculations have been carried out for class (B) molecules, this point appears to have been missed. Almost all NPSO calculations have been carried out for class (C) molecules which will be discussed later. Formal charges and distortion of centroids from pure LDQ theory An important feature of Linnett’s description of double quartet theory is the use of formal charges. The relationship of this feature with the centroids of charge is not discussed by Leroy and co-workers. Related to this is the question of how tetrahedral are the quartets and the problem of socalled L-strain [ 25,261. Nitric oxide is represented by X

X

0

-N-O----

Dividing the electrons the formal charges N-l/2

in the bond region equally

between

the atoms gives

O +1/2

These are contrary charges which are N +0.1566 The positions

to both electronegativity

considerations

and the SERHF

0 -0.1566 of the centroids

of charge in nitric oxide are shown in Table 2.

TABLE 2 Results for nitric oxide (R = 115 pm)’

r, o-N_-__0

r, =40 rz =.74 rB = 50 r, =33

01 = 64.5 e* = 17.7 Be = 26.5 8, = 64.0

ar (pm), B (degrees).

rl = l-1 = r3 = r, =

33 77 49 28

x,OOi _-mm

8, = 18.7 e* = 30.5

0-o

r,

325

While the LDQ theory is clearly shown, there are two distortions from pure tetrahedral quartets equally shared. The “bond” electrons move closer to both the internuclear axis and to the more electronegative oxygen atom. The outer electrons also are closer to the nuclei than the bond electrons. The results for oxygen in Table 1 show the same features. The outer electrons are closer to the nuclei and the inner electrons move closer to the internuclear axis. The two quartets of (Y spin sharing a vertex are close to tetrahedral but the two sets of /_3spin sharing a face have a much contracted shared face. These features appear in all calculations on diatomic species. From the nitric oxide result we can conclude that the pure LDQ picture can distort to overcome unfavourable formal charges. In other cases however the quartets can not distort sufficiently and the LDQ pictures is not found. Some examples are: (i)

FO localises to X

/

0

-F\\

rather than X

\ X

0 O-F=

(LDQ)

Charges are

F +0.5 -0.122 0

0

J-JDQ

SERHF A (ii)

-0.5 +0.122 0

This example, without the charges, agrees with Leroy [14] . NF localises to

rather than X

X

,”

N

F

;

(LDQ)

Charges are

LDQ

UHF A

N -1.0 +0.221 0

F +l.O -0.221 0

(iii) FO+ and ClO’ similarly localise to X

X

0

/

-Ft

(4

326

Charges are FO’ 0

0.0

LDQ

UHF A

c10+ 0 0.0 +0.283 +l.O

F +l.O +0.199 0.0

+0.801 +l.O

Cl +l.O +0.717 0.0

ClO’ is interesting in that the UHF charges are closer to the LDQ formal charges than the formal charges of structure A. This is not the case in FO’. However, in both cases the outer set of three “pairs” on the F (or Cl) are not spatially close. One set in fact lies just on the 0 side of F (or Cl), while the other set lies just on the outer side of F (or Cl). In Table 3, results are presented for some diatomics isovalent with 02, S2 or SO. The set 02, S2 and SO, not surprisingly, fit the LDQ picture. The set NF, PF, NC1 and PC1 do not, although NC1 might be expected to do so. The set NO-, PO-, NS- and PS- fit the LDQ picture except for NS- which, from a charge point-of-view, might be expected to do so more than the first two. The set OF’, SF+, OCI’ and SCl’ do not fit the LDQ picture, although OCl+ might be expected to do so. The conclusion from this is that formal charges are not a good basis for deciding whether a molecule or ion can be represented by a LDQ structure. A calculation of the centroids of charge of localised orbitals will however d.ecide the matter. In polyatomic species LDQ structures separate electrons in some bonds where this appears to decrease bonding. In &HZ, for example, the following spin sets would be predicted H

H X x X

H

X c

x

c

x

cl

spin

X

H

P spin

H’

The “pair” between C and H is not however situated optimally. The /I spin electron lies on the C-H axis, but the 3 o! spin electrons on the H side of C

327 TABLE 3 Diatomic molecules isovalent with 0, A

B

0, S*

so

NF PF NC1 PC1 NOPONSPs-

Formal LDQ charge

UHF charge

A-B

0

0.0

Is structure LDQ?

YES

0

0

S

0.0 +0.338

0

-0.338

A-1

B +1

N P N P

+0.221 +0.391 -0.518 +0.230

F F Cl Cl

-0.221 -0.391 +0.518 -0.230

A-

B 0

N P N P

-0.366 -0.344 -0.746 -0.957

0 0 S s

-0.684 -0.656 -0.254 -0.043

F F Cl Cl

+0.199 -0.105 +0.717 +0.342

-1

OF+ SF+ OCI’ SCl’

, S, or SO

A-B 0

+1

0 S 0 S

+0.801 +1.105 +0.283 +0.658

> NO 1 YES YES NO YES NO 1

triagonally. This situation has been called L-strain by Firestone [25, 261. The case of CzH; has been discussed by Langler et al. [4 3, who misinterpret LDG theory and by Jensen [2’7], who gives a correct formulation of the problem. Using the MIDI-3 basis without diffuse orbitals, the UHF localised orbitals give an interesting structure. The p spin electrons localise exactly as LDQ theory predicts. Five of the a! spin electrons localise in the bond regions and the remaining two are essentially in pure p orbit&, one on each carbon atom. This is not pure LDQ theory but it can still be represented as are arranged

H\

/H

0

cx\H

HlXC

The “pairs” in the C-H bonds are close together, with the p spin centroid slightly nearer to the H atom than the (Y spin centroid. Isoelectronic with &Hi (and 0;) is HO2 . This localises to X X

0 o-

/ OLH

but the odd p spin electron is very close to the OH oxygen atom. In FOz this electron moves over to give

328

Also isoelectronic with 0; is H,NO, a prototype for the stable RzNO radicals. This is discussed by Leroy [16, 171. The centroids give the LDQ structure

H’I:oO

X

H’

but the (Y spin centroid on N is that of a pure p orbital on N allowing the electrons in the N-H bonds to pair up. From these calculations we can conclude that L-strain is indeed a problem and that many structures will not be the pure LDQ formulae in order to allow pairing in bonds. In many cases, however the resulting structure can still be represented by LDQ formulae. This is just another example of the LDQ theory quartets significantly distorting from a pure tetrahedral arrangement. In conclusion, for open shell systems where LDQ theory has been found useful, the centroids of charge of local&d orbitals match the Linnett picture but there are significant distortions from pure tetrahedral arrangements drawing the electrons more closely into bond regions. CLOSED SHELL SYSTEMS

Closed shell systems of class (C) where a single LDQ structure replaces two resonance structures are important. It is here that NPSO wave functions are superior to both molecular orbital and valence bond wave functions. In these cases the idea of centroids of charge of localised orbitals can still be applied but in a more difficult manner. The wave function is the familiar closed shell function * =

[91Q

@2P**..9n~

@“PI

The set of orbitals $J~. . . 9, can be localised by a unitary 8=LqJ

(7) transformation (8)

Newton and Switkes studied localisation of molecular orbitals of class (C) molecules [28]. They found that the non-unique unitary transformation L could be chosen to give localised orbitals matching either of the two resonance structures. For benzene, for example, L can be chosen to give one Kekule structure or it can be chosen to give the other. Thus 8’ = L’ 4 e2 = L2 f$

(9) (16)

where the 0: match one resonance structure and the 0; match the other. If we localise the (Yspin orbitals by L’ and the /3 spin orbit& by L2 , the wave function is transformed to \k =

[e:ole:(u...et,ae:Pe~p...e2,P]

(11)

Equation (11) appears strange but it is identical to eqn. (7). What appears to be a “different orbitals for different spins” function is simply a different way of writing the closed shell function. The centroids of charge of the orbitals 0 i’ and 8 f can be identified with the “dots” and “crosses” of LDQ theory, since one set looks like one resonance structure and the other set looks like the second resonance structure. Thus in principle the LDQ structure is identical to the molecular orbital picture, not only for open shell systems of class (B), but for closed shell systems of class (C). However there are two problems. Firstly the localised orbit& will not be as well localised as those for class (B) systems. In benzene, for example, the localised orbitals for one Kekule structure will give a uniform electron density around the ring. The symmetry of the electron density is not reduced from DC,, to Dsh. Secondly, the method is computationally difficult and it has not yet been implemented within the Foster and Boys algorithm. Equation (11) gives an interesting insight into the NPSO wave functions for class (C) molecules. These are built from “bond orbitals” and “atomic orbitals”. These would arise from truncation and modification of the sets 8 ’ and 6 2. However variation of set f3’ in a different manner to variation of set e2 gives a wave function, formally similar to eqn. (ll), but one which is no longer identical to eqn. (7). Indeed it can no longer be transformed to a function formally similar to eqn. (7). This variation is not allowed within the Hartree-Fock approach. The new NPSO wave function for class (C) molecules, unlike the class (B) case, is not necessarily inferior to the molecular orbital wave function. Indeed since it separates the electrons spatially it may well be superior as was found by Linnett and co-workers [5-lo]. The eqn. (11) wave function is an eigenfunction of the S2 operator. The NPSO wave function arising directly by truncation and modification of eqn. (11) is not. A multidimensional function must be generated, for example, by a projection operator technique. The simplest illustration of this approach is not a NPSO wave function but the Coulson-Fischer [29] function for hydrogen. The simple MO function is

\k =

[ha +

xb)&

(xa +

(12)

xb)Pl

where xa and Xb are 1s functions on the two atoms. Varying the two orbitals in a different manner, followed by expansion to two determinants, leads to the Coulson-Fischer function *

=

[h xa +

xb)a!(xa

Optimisation of resonance function, from the 1s basis. Fock approach and

+ bb)fll

+

[ha +

hxb)a(kia

+

xb)fll

(13)

h gives a function equivalent to the ionic--covalent or in modern terminology, full configuration interaction This variation is of course not allowed by the Hartreeit gives a lower energy.

330 CONCLUSION

The use of localised orbitals, for the most common type of both open and closed shell systems where Linnett’s approach has been found useful, clarifies the close relationship between molecular orbital and LDQ theory. Furthermore it appears to be the only sound basis for determining whether the LDQ approach is appropriate for a particular system and it clarifies the question of distortion of Linnett’s quartets from a purely tetrahedral arrangement. The use of formal charges, as advocated by Linnett, to determine the appropriateness of LDQ structures is shown in general to be unreliable. ACKNOWLEDGEMENTS

An annotated bibliography of work arising from Linnett’s original LDQ proposal is available on request from the author. The assistance of the computer centres of the Universities of Lancaster (U.K.) and Papua New Guinea is acknowledged. REFERENCES 1 J. W. Linnett, The Electronic Structure of Molecules - A New Approach, Methuen, London, 1966. 2 J. W. Linnett, J. Am. Chem. Sot., 83 (1961) 2643. 3 W. F. Luder, The Electron Repulsion Theory of the Chemical Bond, Reinhold, New York, 1967. 4 R. F. Langler, J. E. Trenhohn and J. S. Wasson, Can. J. Chem., 58 (1980) 780. 5 D. M. Hirst and J. W. Linnett, J. Chem. Sot., 1962 3844. 6 D. M. Hirst, Mol. Phys., 31 (1976) 1511;Mol. Phys., 33 (1977) 369. 7 P. B. Empedocles and J. W. Linnett, Proc. Chem. Sot., (1963) 303. 8 D. M. Hirst, J. Chem. Sot., Faraday Trans. 2,73 (1977) 443. 9 B. J. Duke and J. W. Linnett, Trans. Faraday Sot., 62 (1966) 2955. 10 D. Gould and J. W. Linnett, Trans. Faraday Sot., 59 (1963) 1001. 11 D. M. Hirst and M. Linington, Theor. Chim. Acta (Berlin), 16 (1970) 55. 12 D. Peeters and G. Leroy, Bull. Sot. Chim. Belg., 86 (1977) 833. 13 G. Leroy, D. Peeters, A. Depius and M. Tihange, Nouveau J. Chim., 3 (1979) 213. 14 G. Leroy, in I. G. Csizmadia and R. Daudel (Eds.), Computational Theoretical Organic Chemistry, 1981, p. 253. 15 G. Leroy, J. Mol. Struct. (Theochem), 93 (1983) 175. 16 G. Leroy, J. Mol. Struct. (Theochem), 103 (1983) 45. 17 G. Leroy, Adv. Quantum Chem., 17 (1985) 1. 18 G. Leroy, D. Peetem and M. Tihange, J. Mol. Struct. (Theochem), 123 (1985) 243. 19 J. M. Foster and S. F. Boys, Rev. Mod. Phys., 32 (1960) 300. 20 J. S. Pople and K. Nesbet, J. Chem. Phys., 22 (1954) 571. 21 R. Daudel, G. Leroy, D. Peeters and M. Sana, Quantum Chemistry, Wiley, New York, 1983. 22 H. Tatawaki and S. Huzinaga, J. Comput. Chem., l(l980) 205. 23 T. Sakae, H. Tatawaki and S. Huzinaga, J. Comput. Chem., 2 (1981) 100. 24 M. F. Guest and V. R. Saunders, Mol. Phys., 28 (1974) 819. 25 R. A. Firestone, Tetrahedron Lett., (1968) 971. 26 R. A. Firestone, J. Org. Chem., 34 (1969) 2621. 27 W. B. Jensen, Can. J. Chem., 59 (1981) 807. 28 M. D. Newton and E. Switkes, J. Chem. Phys., 54 (1971) 3179. 29 C. A. Co&on and I. Fischer, Philos. Mag., 40 (1949) 396.