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An interpretation of chemical bonding in a molecular orbital wavefunction
Journal of Molecular Structure (Theochem), 169 (1988) 211-232 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
211
AN INTERPRETATION OF CHEMICAL BONDING IN A MOLECULAR ORBITAL WAVEFUNCTION*
ALICE CHUNG-PHILLIPS
Department of Chemistry Miami University, Oxford, OH 45056 (U.S.A.) (Received 17 November 1987)
ABSTRACT Localized atomic orbit& (LAOS) are atomic hybrid-like orbital8 derived from self-consistentfield molecular orbitals (MOs); their directional characters offer a lucid interpretation of localized electronic properties. Electronic populations and energies associated with individual LAOS and pairs of LAOS may be combined to yield the corresponding properties for the localized chemical orbitals ( LCOs ) . These properties are particularly useful for the study of bonded and nonbonded interactions within a molecule. In this paper the ab initio LAO methodology is both reviewed and updated. For the purpose of demonstrating different facets of this methodology, details of deriving LAO (and LCO) populations and energies from a minimal A0 basis (STO-3G) are shown for the water (H,O) molecule. Preliminary results from extending the minimal to an extended A0 basis of split-valence (43lG) for H,O are also reported.
INTRODUCTION
The nature of the chemical bond has been indisputably a subject of fundamental interest throughout the history of chemistry. The octet rule postulated by Lewis based on electron-pair bonds [ 1] and the average bond energies deduced by Pauling from thermochemical data [2] are still in popular use. Later developments follow the mainstream of quantum mechanics in the study of electronic structures of molecules. In constructing the electronic wavefunction for a complex molecule, the molecular orbital (MO) method is usually preferred to the valence bond (VB) method. The self-consistent-field (SCF) procedure of deriving MOs as linear combinations of atomic orbitals (LCAOs), standardized by Roothaan [3] in the early 19508, yields quantitative information on the electronic energy of the molecule as a whole and the energies of individual MOs. Subsequent application of the Mulliken population analysis [ 4 J to the LCAO MOs results in electron population terms for each A0 (or atom) and between each pair of AOs *Dedicated to Professor Linus Pauling.
0166-1260/66/$03.50
0 1988 Elsevier Science Publishers B.V.
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(or atoms) in the molecule. These computed energy and population quantities are very useful in the interpretation of spectroscopic data and photochemical properties of molecules. In regard to the chemists’ concept of chemical bonds [ 1,2] as electron pairs localized in different regions of the molecule, the conventional MO method is inadequate. The problem lies in the fact that MOs are delocalized orbitals which do not convey the concept of directed valence. This problem imposes upon chemists a conceptual barrier in understanding the properties of electron-pair bonds and a practical difficulty in estimating the extent to which nonbonded interactions (those between electron pairs from different localities) determine the relative stabilities of structural or conformational isomers. In the SCF MO LCAO treatment of molecules the typical quantum chemical approach to the study of isomeric species is to use the calculated total energies to establish their relative stabilities and to use populations to provide some measure of general intramolecular interactions. Despite this restrictive mode of approach, great strides have been made in the understanding of molecular stabilities from using the semiempirical complete and intermediate neglect of differential overlap methods (e.g. CNDO and INDO of Pople [ 51, MIND0 of Dewar [ 61, PCILO-CNDO of Malrieu [ 71 and their respective co-workers) and the ab initio MO methods with Gaussian-type basis sets (e.g. STO-3G, 431G, .... and MP2/6-31G* methods of Pople and co-workers [8-lo]. ’ To make the MOs more meaningful in the interpretation of chemical bonds it has been known for some time that the delocalized MOs for a closed-shell configuration can be changed into a set of localized MOs (LMOs) [ll] by means of an orthogonal transformation. The most familiar LMO procedure, that of Edmiston and Ruedenberg [ 12, 131, is based on the minimization of the total interorbital (inter-MO) charge-cloud repulsion in the molecule. This method has been successfully applied within both the ab initio and semiempirical frameworks [ 141. In the past decade the major research efforts of this laboratory have been centered on the study of covalent bonds and their associated properties extracted from molecular orbitals. Initially, a theoretical procedure was formulated to partition the binding energy calculated from a semiempirical MO method into two-atom terms [ 151 for the purpose of finding quantum chemical quantities that can be directly related to the kind of bonded and nonbonded interactions discussed traditionally [ 161. Applications of this procedure to improved CNDO wavefunctions for saturated hydrocarbons, amines, alcohols, and ethers [ 171 produced “bond energies” comparable to those obtained from thermochemical data [ 181. More recently the ab initio method of localized atomic orbitals (LAOS) for atoms in molecules was proposed by Aufderheide [ 191; The LAO method implements the standard SCF MO procedure with an orthogonal transformation of the (orthogonal) A0 basis on an atom to the LAO basis on the same atom.
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The transformation is based on the maximization of the total interorbital (inter-AO) charge-cloud exchange energy for the given atom in the molecule. The resulting LAOS are hybrid-like core, bonding, and lone-pair orbitals. The LAOS on the same atom are orthogonal, but those on different atoms are nonorthogonal. After the transformation, the original MOs may be expressed directly as linear combinations of LAOS. This enables a natural partition of the calculated electronic population and energy for the molecule into population and energy terms associated with individual LAOS and pairs of LAOS. Applications to small molecules [ 20, 211, using STO-3G wavefunctions, have given strong evidence that the model is well suited to the study of general intramolecular interactions in complex molecules. The LAO algorithm used in connection with STO-3G A0 basis sets from GAUSSIAN 70 [22] has also been made publicly accessible through the Quantum Chemistry Program Exchange [23, 241. In the intervening years the LAO method has been modified and extended in order to develop the model into a practical and versatile theoretical tool for in-depth analyses of molecular electronic structures [ 251. Specifically, the original LAO formulation has been extended from closed-shell saturated molecules to open-shell and conjugated systems for a broader range of applications, and from the minimal A0 basis (STO-3G) to an extended A0 basis of split valence (4-31G) for more accurate studies of energies. Additional efforts have been made in the design of the extended LAO computer program [ 251 for a reduction of computation time through selective A0 (or atom) localization. In this paper, preliminary findings in these areas will be reported. THE ORIGINAL LAO FORMULATION
The criterion for the LAO localization is nonarbitrary; it is a natural extension of the principle of localized orbital correspondence between the LMOs in a molecule [ 12,131 and the LAOS on the free atoms [ 191. The resulting LAOS ‘have been shown to be physically meaningful [20] and transferable among molecules of similar bonding environments [ 211. To facilitate a discussion of the extension of this method, the original formulation will be summarized with conventional notations. Derivation of LAOS
For a molecule with 2n electrons forming a closed-shell configuration, the approximate electronic wavefunction Y may be written as a single Slater determinant over n occupied pairs of spin orbitals, ~~vicx and w&3[ 31 Y(1,2,...2n)=(y/,(l)a(l)W,(2)8(2)...cy,(2n-l)a(2n-l)cy,(2n)B(2n)I (1)
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In the LCAO scheme the spatial orbitals cyare usually the SCF MOs expanded from a basis set of real AOs $
y=@C
or
Yi=C
cpi
#p
P
(2)
where C contains the linear coefficients. (Indices i, j for MOs and p, V,1, rs for AOs are used.) The electronic energy E may be expressed directly in terms of certain A0 quantities (3) where Pw and HP are the density and core matrix elements, and @~/no) denotes a two-electron interaction integral [ 5 1. The electron distribution in the molecule may be estimated by means of the Mulliken population analysis [ 41. The total number of electrons is decomposed into a sum of terms’
where SW is the overlap integral. The term qp represents the electron charge contributed by the A0 charge density (CD ) , Q&#,,. Suppose now a similar decomposition of the energy E is desired for providing a description of the “energy distribution”. Analogous to eqn. (4)
where Ew must be properly defined so as to represent the energy distribution associated with the A0 CD qJ,&. A close scrutiny of the conventional expression for E as given in eqn. (3), however, reveals that the integral (puo/lv) in the exchange energy term refers to an interaction between A0 CDs #,,@,and &J,, neither of which relates to &,A. To highlight the A0 CD of interest, $,,$,,, the alternative expression for E is used [ 191
Using this new expression the partitioning of E into contributions from individual A0 densities $,&,, as prescribed in eqn. (5) can be realized with
Ew=&vfb + C C (4P&o - i PJ’av ) W/J-a) Note that this new form of expressing the exchange energy offers a means for the localization of AOs as described in eqn. (10) below. The usefulness of eqns. (4) and (5) becomes more evident if, instead of the A0 basis #, a different basis x is used such that x reflects the property of di-
215
rected valence. In this study x is specifically the set of hybrid-like LAOS related to # by an orthogonal transformation T p$T
or
while keeping the MOs invariant ry=@C=xD
and
D=‘i’C
(9)
(Indices a, b, c, d are used for LAOS.) T is a block-diagonal matrix containing one block TA for each atom A in the molecule. 9 is the transpose of T. TA is obtained by an iterative process which maximizes the exchange function LA deduced from eqn. (6). LA evaluated in the A0 basis, #, is I
(10) whereas its maximum value is reached in the LAO basis x (11)
Note that the summation over index ,u or a is over orbitals on atom A only, whereas the summations over indices 2 and o must include all orbitals in the molecule. The resulting TA is orthogonal as the AOs on each atom have been made orthogonal initially. TA defines the composition of the LAOS on atom A in terms of the AOs on the same atom. Using T, the transformation from the A0 basis to the LAO basis for the oneelectron quantity M is carried out as M(X) =T M(#)T
(12)
where M represents the density matrix P, the overlap matrix S, the core-Hamiltonian matrix H, or one of the Cartesian moment matrices X, Y, and 2. As for the two-electron integrals the time-consuming task of full transformation is avoided by using, for example, the mixed-basis quantities P,, Pb, and (au/ Aa) as in eqn. (11) which, upon multiplication and then summation over all A0 charge densities @J&,,become equivalent to using pure LAO quantities Pa,, Pbaand (aa/bc) over all LAO charge densities xbxc. The diagonal elements of X(x), Y (2) , and Z (x) are next used to determine the direction and mean distance r(a) of an electron in the LAO xa relative to the atomic nucleus. From these directions, the angle between two LAOS X~and xb on the same atom, 0( a$), can be deduced. These terms provide a geometrical description of the LAOS.
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LAO (or LCO) electronic populations and energies
To study localized properties, a knowledge of electronic population and energy distributions over atomic-core, lone-pair, bonding, and other chemically significant regions is essential. Thus, it is important to partition the total electronic charge, 2n, and the electronic energy, E, in the LAO basis
(134 where qab = Pab Sab
(13b) (14a)
(14b) Transformations described in, and subsequent to, eqn. (12) are used in the calculation of the one- and two-electron quantities in the above two equations. The LAO population and energy terms (q& and Eab, respectively) furnish a convenient set of building blocks for constructing interatomic bonding terms by simple summations. Proper groupings of certain valence LAOS on different atoms will not only illuminate the bonding features but also reduce the number of significant terms for subsequent analyses. These groupings, plus the core and lone-pair LAOS, may be called the localized chemical orbitals (LCOs) . (Indices y, 6 are used for LCO terms.) Example: the Hz0 molecule
Presently it is instructive to use a numerical example for clarifying notations and the major steps in the LAO procedure. For this purpose the water (H,O) molecule is selected owing to its simplicity and importance in chemistry. Furthermore, a wavefunction expanded from a minimal A0 basis, STO-3G, is chosen due to its availability [ 8,221. It should be emphasized that the numerical results from this exercise will not be sufficiently accurate because of the nature of the wavefunction. If the wavefunction has taken into account electron correlation [lo]and is also constructed from an extended A0 basis including polarization functions, then the results can be expected to approach chemical accuracy. Hz0 has the experimental geometry of 0.957 A for the bond length and 104.5’ for the bond angle, which may be compared to the STO-3G optimum geometry
217
of 0.990 A and 100.0” [8]. Experimental and STO-3G data pertinent to H,O are tabulated in Tables 1 and 2. In Hz0 there are seven AOs: five on the oxygen atom (Is, 2s, 2px, 2py, 2pz), and one on each hydrogen atom (Is). (Note that the 1s and 2s orbitals on oxygen have been made mutually orthogonal subsequent to the STO-3G calculation.) The valence AOs include the oxygen 2s, 2px, 2py, and 2p.z orbitals plus the two hydrogen 1s orbitals designated as hl and h2. These six valence AOs are schematically represented in Fig. 1 (top). From the SCF MOs, 4 valence LAOS for oxygen are derived from the 4 valence AOs with the frozen-atomic-cores approximation (vide infra) via eqns. ( 10) and ( 11). These are the lone-pair LAOS, Pl and 1/2,and the bonding LAOS, bl and b2 a1 =a(2s) -b(2px)
+c(2pz)
Q2=a(2s) -b(2px) -c(2pz) bl=a(2s) +b(2px) +c(2py) b2=a(2s)
(15)
+b(2px) -c(2py)
where a= 0.58565, b = 0.39624 and c = 0.70711. The associated average electron distances and interorbital angles related to the LAOS are: r(Q1) = 0.325 A, r(bl)=0.249 A, 13($?l,Q2)=121.5”, kQl,bl)=108.2”, and 8(bl,b2)=100.7”. The remaining AOs are: i, representing the oxygen 1s core orbital; and hl and h2, the hydrogen 1s orbitals. These three AOs become LAOS automatically since they require no localization. The valence LAOS are displayed in Fig. 1 (center). The partitioned charges, qab,and energies, Ea6, for the 7 LAOS (i, Ql, 112,bl, b2, hl, h2) based on eqns. (13) and (14) are presented in Table 3. Note that the tabulated values Ma,,, where M=q or E, are expressed as M(a) = Ma, for TABLE 1 The Hz0 molecule: STO-3G values versus experimental values8*b Quantity
Experimental value
STO-3G value
Geometry t(O-H) (A) B(HOH) (deg.) Dipole moment (D ) Atomization enen@ (kcal mol- ’ ) O-H bond energyd (kcal mol-’ )
0.957 104.5 1.846 232.2 116
0.990 100.0 1.51 143.4 72
“STO-3G values for the optimum geometry were taken from ref. 26. Other values are calculated for this work based on the STO-3G optimum geometry. bExperimental values were taken from: for geometry, ref. 27; for dipole moment, ref. 28; for atomization energy, refs. 29,30. “Energy for the reaction, H,O (g) +2H (g) +0(g) at 0 K and corrected for zero point energy. %&en as onehalf of the atomization energy.
218 TABLE 2 Pertinent data (a.u.) from STO-3G calculations for H,O A. Optimum geometry (in Cartesian coordinates) Y 0 f.6 0.0 f.0 Hl 1.20257 1.43317 0.0 H2 1.20257 - 1.43317 0.0 &ZB
B. Nuclear repulsion: 5 g 1;\g=8.96107 ‘OZH ‘*’ -=1.87068 rOH
~427610 *
z,zH=~=O_3488f$ rHH
2.86633
C. Free atom energies: F Ei= - 74.73731 0: E= - 73.86415 H: E = -0.46658 D. Electronic energy: E = - 83.86698 Et = E + (nuclear repulsion) Total energy: = - 74.96591 Binding energy: Eb = Et- (free atom energies) = -0.22860
Fig. 1. Schematic representations of the valence orbitals of H,O: top, the atomic orbit& (AO); center, the localized atomic orbit& (LAOS); bottom, the localized chemical orbit& (LCOs ) .
219 TABLE 3 The H,O molecule: LAO electronic population and energy terms (STG-3G)B*b (i)
(Ql)
(bl)
(hl)
- 52.945 2.005
- 2.107 8.374
- 0.802 3.250
- 0.627 1.731
(i,Ql)
(W2)
(bl,b2)
(hl,h2)
-0.158 0.0
-0.052 0.0
-0.091 0.333
(Pl,bl)
(blhl)
- 0.0 0.089
- 0.692 2.737
(Chl)
(blh2)
-0.076 0.312
- 0.030 0.134
- 0.082 0.0 (i&l) - 0.0 0.022 (i&l) - 0.004 0.092
“Energies are given in a.u. Notations are defined in text. Terms equivalent by spatial symmetry to those listed are not shown. The same convention is followed for Tables 4-10. bDefinitions:N=~q(a)+~~q(a,b)andE=~E(a)+~~E(a,b)whereNisthetotalnumber of electrons an: E is the:konic energy. Vkes for tii*ppulation terms, q(u) and q(a,b), and energy terms, E(a) and E(a,b) , based on the localized atomic orbit.& L, xb, am listed under theLAOlabels (a) and (a,b).NotethatforM=qorE,M(~)=M,andM(a,b)=M~~+&,in relation to eqns. (13) and (14).
TABLE 4 The Hz0 molecule: LCO electronic population and energy terms ( STO-3G)“*b
ti)
(al)
(01)
:
- 52.945 2.005
2.107 - 8.374
2.120 - 7.718
:
(i,Ql) -0.082 0.0
(QW) 0.0 -0.158
(ol@) -0.150 0.550
(i,ol)
(QlPl) - 0.076 0.223
z
- 0.004 0.070
“See footnote a of Table 3. bDefinitions: N= C q(y) + TcT q (y,6) and E = C E( y) + T<& E( y,6) Y Y where N is the total number of electron and E is the electronic energy. Values for the population terms, q(y) and q(yJ), and energy terms, E(y) and E(yJ), based on the localized chemical orbitals, are listed under the LCO labels (y) and (~,a). Note that for ME q or E, M(y) = M, and MY,& =A&+ i+&
a= b and M(a,b) = M,,b + Mb,, for a # b. The terms equivalent by spatial symmetry to those shown (for (Q2), (b2), .... (b2,hl) ) have been omitted from the list. Summation of the 28 LAO terms (7 X 8/2) gives 10 electrons for the charge and -83.867 au. for the energy. Grouping each bonding LAO on oxygen with the hydrogen Is orbital in its directed path leads to a “o-bond orbital”. Symbolically, the two bond orbitals are: al = bl + hl and a2 = b2 + h2, where + defines how a calculated M(a) or M(a,b) should be summed. Thus, the population in al is: q(a1) =q(bl) +q(bl,hl) +q(hl). Likewise, the overlap population between 01 and a2 is: q(al,o2) =q(bl,b2) +q(bl,h2) +q(b2,hl) +q(hl,h2). Applying this procedure, the 4 LAOS (bl, b2, hl, h2) are, in essence, replaced by 2 bondorbitals (al, o2), making a total of 5 LCOs (i, Ql, Q2, al, 02). The 4 valence LCOs (al, Q2, 01, o2) are symbolically represented in Fig. 1 (bottom). The LCO population and energy terms, qysand I&, are given in Table 4. Note that the LCO terms for (i), (al), (i,Ql), and (Ql,Q2) are not different from the previous LAO terms. There are a total of 15 LCO terms (5 x 6/2), a reduction of nearly one half compared to the number of the LAO terms. The calculated values for Hz0 as exhibited here are illustrative of the kind of numbers sought by quantum chemists for quantifying popular chemical concepts and trends. For instance, the average distance of an electron in a lonepair orbital on oxygen (r (Ql) or F (as) ) and their relative orientation [ 8( Ql,Q2)] influence the geometry of hydrogen-bonded clusters in’ an aqueous solution. The charges and energies in bonded regions [ (al ), (02) ] and nonbonded regions ((Ql,ol), (Ill,&!), (Q2,01), (112&Z),(al,o2)),versus those relatedto the lone pairs [ (Ql), (Q2), (Ql,Q2)1, affect the O-H bond strength and the geometry of H,O. EXTENSIONS OF THE LAO METHOD
To make the LAO method a practical and versatile theoretical tool, many fundamental problems need to be resolved. In the following, preliminary reports are given for advances made in certain areas: applications for chemical systems other than the closed-shell saturated compounds (first and second sections), reduction of computing time to allow for treating molecules with large numbers of electrons (second section), extending the A0 basis to a splitvalence basis for better accuracy (third section ), and the partitioning of total and binding energies for studies of bonded and nonbonded interactions (fourth section). Open-shell configuration
For an open shell there are two separate density matrices, P (@)”and P (qb)p for the cy and /3 electrons, respectively. The full density matrix, P (#), is the sum of these two
221
To derive the LAOS we propose to carry out the maximization of LA using P (@) in eqn. (10) and arrive at a single T which is to serve both cy and p electrons. For subsequent calculations of LAO quantities we will apply T to P(@)“, P(@)? P(#), and other A0 quantities. Finally, the qaband Eabterms are obtained from eqns. (13) and (14) except for the replacement of (l/ 4)P,,PAb in the exchange energy term by ( 1/8)Pa0LYPAb(Y+ ( 1/8)P,BPA68. This open-shell procedure has already been incorporated in the LAO computer program [24]. Selective AOlatom localization
The AOs on atom A may be separated into subsets (p), (v), etc. One may choose to localize a particular subset (p). To do so, one simply excludes all AOs on A not in (p) in the first summation for LA(#) in eqn. (lo), but keeps the other two summations (over 1 and o) intact (The latter summations over all A0 charge densities #,&, accounts for the full potential field of the molecule as determined by the SCF MOs.) One form of this approach is the approximation of frozen atomic cores (FACs), which is an atomic core-valence separation based on energy. The core AOs are assumed to be localized sufficiently to be taken as the core LAOS at the outset. This approximation effects considerable savings in computational efforts and also produces sufficiently accurate results. For Hz0 (vide supra: Example: the H,O molecule), a six-fold time reduction is achieved through FACs with a difference of 0.005 electron in q(!?l) or q(b1) from the complete treatment [ 201. The FACs treatment has been well tested [ 241. Another familiar example of this approach (based on symmetry) is the 0-7~ separation in a conjugated molecule, where the n-AOs are equated to the JZLAOS at the outset, leaving the o-AOs to be localized. For each carbon atom in benzene (C&H,) for example, the 2pz A0 perpendicular to the molecular plane is treated as a K-LAO, while the 2s, 2px, and 2py AOs are localized to yield the three bonding O-LAOS. Analogously, the 2pz A0 for a Hz0 molecule lying in the xy plane may be made into a lone-pair LAO at the outset, yielding a description of two nonequivalent lone pairs in H,O. To test this bonding scheme for H20, the STO-3G wavefunction (see, Example: the H20 molecule) is subjected to this selective A0 localization; i.e. excluding (or freezing) the 2pz A0 from localization. In other words, 2pz is made into the lone-pair LAO, ll~, at the outset. The transformation for the 3 valence AOs on oxygen to the new
222 TABLE 5 The Hz0 molecule: LCO electronic population and energy terms with selected A0 localization (STO-3G)’
(i)
(Qlr)
(Qu)
(al)
:
- 52.945 2.005
2.000 -7.568
2.097 - 8.327
2.154 - 7.954
(i,Qn) -0.025 0.0
(Qn,Qa) 0.0 -0.112
(QWl )
:
-0.051 0.155
(&a2) -0.099 0.372
:
(i&Q -0.016 0.0
(Qwl) 0.0 -0.059
:
(Cal ) --0.031 0.002
“See footnote a of Table 3 and definitions (footnote b ) in Table 4.
lone-pair LAO, Ilo, and the bonding LAOS, bl and b2 is given as Qa=a’ (2s) +b’ (2p3c) bl=a(2s)+b(2px)
+c(2py)
b2=a(2s)+b(2px)
-c(2py)
(16)
where a’ = 0.66328, b’ = -0.73016, a=0.51630, b=0.48315 and c=O.70711. The associated average electron distances and interorbital angles related to the new LAOS are: r(Qa)=0.342 A, r(bl)=0.303 A, 8(Qa,bl)=124.3”, and 8(bl,b2) = 111.3”. The LCO population and energy terms based on this set of LAOS are shown in Table 5. The atoms in a molecule may be grouped according to symmetry. In each group, one may select only one atom for maximization of LA and then relate the resulting LAOS to all other symmetry-equivalent atoms via orthogonal transformations. For example, only one carbon atom in benzene needs to be treated explicitly. In fact, a complete freedom exists in the choice of atoms for localization regardless of symmetry. In pyridine, if the “reactivity” of the lone pair on nitrogen is the sole interest, the carbon atoms will not require explicit LAO considerations. This option is particularly convenient and economical for certain comparative studies in related molecules. Extended A0 basis The single-configuration wavefunction spanned by a minimal A0 basis, as employed in the current LAO method, is inadequate for the determination of very accurate relative energies. It is well known that the use of configuration
223
interaction (CI) or some variant, coupled with an extended A0 basis, will improve the wavefunction in this respect. These added features to the original wavefunction would mean the following in deriving the LAOS. The upper limits for all summations over the A0 or LAO indices in the prior equations must be numerically increased. The density matrix elements must now reflect contributions from more than one configuration or weighted occupations of MOs. The cost in computation for these additional features, if drastic measures are not taken to modify the iterative step represented by eqns. (10) and (ll), would render the LAO procedure totally impractical for large systems. The specific LAO treatment of a CI wavefunction and/or the use of polarization functions in the extended basis will be postponed. At present, the interest in practical applications calls for an extended A0 basis containing splitvalence, or double-zeta, orbitals at the level of 4-31G [ 91. A “short-cut” approach which gives reasonable results is described below. Take the HZ0 molecule (vide supra, Example: the Hz0 molecule), again, as an example. For the 4-3lG basis there are 8 valence AOs for oxygen, falling into two sets: (2sI,2pxI, 2pyI,2pzI) of Set I and (2sO,2p~O, 2pyO,2pzO) of Set 0, corresponding to replacing each orbital exponent (zeta) in the minimal A0 basis by a larger (Set I) and a smaller (Set 0) exponent. To obtain preliminary LAOS for the 4-31G calculation for HZ0 (based on the STO-3G optimum geometry as in the Example section), the A0 coefficients for the LAOS from the STO-3G calculation in eqn. (15) may be directly transferred to Set I and Set 0 to yield two sets of 4 LAOS: (QlI, Q21,bl1, b21) of Set I and ($90, P20, b10, b20) of Set 0. To obtain LAO charge and energy terms, the valence LAOS in Set I and Set 0 on oxygen are merged into a single set of “four” orbitals based on transformation properties: {QlI, alO>, (P21, !?20}, (bl1, blO}, and (b21, b20). Adding to these four orbitals the core orbital of oxygen, {i}, and the two hydrogen orbitals, (hl1, h10) and (h21, h20), “seven” LAOS are again obtained as in the case of a minimal A0 basis. In this extended basis, each calculated LAO term contains contributions from components I and 0. Results for the LCO analysis based on the 4-31G basis with this procedure are presented in Table 6. If greater accuracy is desired, the localization procedure may be applied selectively on AOs of the same angular characteristics first, e.g. between 2sI and 2sOto get 2s’ and 2s II, between 2~x1 and 2~x0 to get 2px’ and 2px”, etc. Next, use the A0 coefficients for some “standard” LAOS (e.g. those obtained from STO-3G calculations in eqn. (15) ) as initial input for the iterative LAO derivation on the set with lower energies (2s’) 2px’, 2py’, 2~2’ ) . Finally, apply the transformation matrix thus obtained for the previous set directly to the set of higher energies (2s II,2px”, 2py”, 2~2” ) without going through further iterative LAO derivation. In short, different procedures are available for treating extended A0 sets.
224 TABLE 6 The Hz0 molecule: LCO electronic population and energy terms using an extended A0 basis of split-valence (4-3lG)’
6)
(al)
(al)
:
- 53.295 1.997
2.111 - 8.246
2.096 -7.911
:
(i,Ql) - 0.087 0.0
(QW) 0.0 -0.138
(ol@) -0.136 0.498
;
(i,al) - 0.051 0.0
(QlPl) -0.069 0.180
‘See footnote a of Table 3 and definitions (footnote b ) in Table 4.
The complete treatment, i.e. using all the AOs of a 4-3lG basis in the iterative LAO derivation at once, is unjustified in view of its unreasonable demand on computer resources. The most expedient way is that which has been shown for the H,O molecule. Partitioning of total and binding energies
In order to model energy variations arising from isomeric changes, energy quantities more illustrative than the electronic energy E should be considered. These are the total energy Et (E plus the nuclear repulsion) and the binding energy E,, (Et minus the free-atom energies) in a.u. E,=E+C
&F
(17a)
and &=E,--C
E;;
(17b)
A
(Indices A, B are used for atoms. ) To assess regional contributions to the energy variations, Et and Eb should also be partitioned in the LAO basis yielding &,ab and &,ob analogous to E and Eab in eqn. (14). To do so, unfortunately, calls for arbitrary assumptions with regard to the partitioning of the nuclear repulsion and free-atom energy terms in eqn. (17). For initial trials, the following approaches may be adopted. The free-atom energy in its ground state, E& may be replaced by PA, the energy of the valence state as determined by the bonding environment of A in the molecule, and LIE;, the promotion energy
225
Eb=Et-
c A
(PA-AEi)
(18)
The calculation of E*Ais next carried out for the best estimated valence state. (As an example, for C in CHI, E; corresponds to the ls22s22p2 3P state and E*A,the ls22s2p3 4S state). In the LAO context the valence state is defined by the transformation TA derived for the molecule in question. Accordingly, the charge density obtained for the free-atom valence state, PX, is transformed from the A0 basis to the LAO basis as in eqn. (12) PX(X)=*APZ(#)TA
(19)
Other A0 quantities have already been similarly transformed for the molecular calculation in eqn. (12). Using these transformed quantities the LAO charge and energy terms, qz,aa and E*A& are obtained via eqns. (13) and (14). (Note that q&, = 0 for a # b because S&, = 0. ) In the absence of a better scheme at present, the partitioning of the promotion energy follows a more ad hoc route. A simple equation containing a weighing factor w, may be tested first [ 151 AE+CA
w, AEX
For simplicity, consider the case of FACs (vide supra). Then, for the core LAO (eqn. 21a) and the valence LAO (eqn. 21b), respectively w,= 0
(21a)
and w,=
&z, number of valence electrons on A
(21b)
for which C Aq&a = number of valence electrons on A a over the valence LAO index a for atom A. This approach is based on the assumption that a valence LAO carrying a greater charge has received a greater share of the promotion energy. (For C in CH4, w,= l/4 for each valence LAO.) There is no simple prescription for resolving the nuclear repulsion term, One approach is to split up the nuclear charge on each atom into zAzB/rAB* different LAOS; each LAO receives a certain amount depending on the nature of the LAO. Thus, for ZA on atom A and .& on atom B zA=xAza a
and
(22a)
226
(22b)
ZB=p
(For C in CH4, Z,= 2 for the core orbital and Z,= 1 for each of the bonding LAO, whereas for H, Zb= 1 for the 1s orbital.) The final partition must be in the form
e= c*
F B‘%,bb
EN,~, +
a
implying that the nuclear repulsion will affect only the “diagonal” electronic E, and Ebb, but not the “Off-diagOnai” tk?rIIM, &b and Eba. Specific details for EN,m and EN,bbneed to be worked out for each group of related molecules under study, e.g. molecules in the series CnHzn+z, AH, or H,ABH,, where A, B = C, 0, N, F, etc. To summarize, the partitioning in the LAO basis for Et and E,, energieS,
and
Eb= 5 F %,a6 involves the following approximations: E t,ob
=
‘%b
+
6%.ab
for a and b both on A @Sal
)hb
and E b,ab =Et,ob
-
[E*A,ab-
(W%ik&b
1
(25b)
andforaonAandbonB E t,ab =E&
(264
and E b,ab =
Et,ab
(26b)
The justification for any of these approximations must come from a high degree of success in explaining the physical origins of the observed energy variations in isomeric species [ 141.It is foreseeable that a reasonable scheme for the energy partitioning will materialize from future trial calculations. As a numerical example, the H,O molecule (vide supra) is again used for further demonstration. The nuclear repulsion is handled in a way that will allow the binding energy of a bond orbital 01, Eb (cd), to be lower than that of a lone-pair orbital Ill, Eb (II1) . The details are given next. The 8+ charge on oxygen is partitioned as follows: 2 + for the core LAO and 3/2 + for each of the four valence LAOS. For hydrogen, 1 + is assigned to the
227 TABLE 7 Pertinent data for use in partitioning A. Nuclear repulsiond (i) 2x EN 1.069 B. Free oxygen atom (i) 2.000 : - 51.787
the total and binding energies of Hz0 (STO-3G)“a*c*d
(al)
(bl) (3/2)x
3x 1.604
WI (5/2)~+(1/2)~ 1.511
0.802
(Ql)
(bl)
1.843 - 6.553
1.157 -4.002 (bLb2) 0.097
E
(i,Ql) -0.068
(w!2) -0.017
E
($1) -0.039
(Qlbl) -0.145
‘See Table 3 for references and the text for details. YI’he nuclear repulsion (-8.90107 a.u.) and electronic energy of oxygen ( -‘73.89415 a.u.) given in Table 2 are partitioned into LAO energy terms. The LAO population terms for oxygen are listed for the nonxero terms only. “For the free hydrogenatom, q(h1) =l.OOOand E(h1) = -0.467 a.u. dFrom Table 2 B: z=l/ron=0.53452a.u. andy= l/ruu=0.34888a.u.
TABLE 8 The H,O molecule: LAO total and binding energy terms (ST@3G)”
Et & Et Eb
6)
(al)
(bl)
(hl)
-51.876 - 0.089
-6.770
-2.448
-0.220
W 1
(QW 1
(bl&‘)
-0.158
-0.014
-0.141
-0.051 - 0.045
-0.022 0.016
(i,hl) Et Eb
1.554
-0.082
fib1 1 Et Eb
-0.217
0.092 0.092
U&b11 - 0.089 0.056
(Ql,hl)
(bl,hl) -2.737 -2.737
(bl,hZ)
0.312
0.134
0.312
0.134
“See Table 3 for references and the text for details.
0.247
W&2) 0.333 0.333
228 TABLE 9 The H,O molecule: LCO total and binding energy terms (STO-3G)’
6) Et &
- 53.876 - 0.089
(al)
-6.770 -0.217
(01)
Wl)
W,L2)
(al@)
-0.082 -0.014
Et &I
-0.158 -0.141
(i,ol) E!J
0.550 0.646
W,al)
0.070 0.109
Et
- 5.406 - 0.936
0.223 0.368
*See Table 4 for references and the text for details.
hl or h2 orbital with no ambiguity. Let X= l/roH and y= l/~r..m, in a.u.. The repulsions between 1+ charge in hl with the 8+ charge in the oxygen LAOS and with the 1+ charge in h2 can be broken down in the LAO representation as (hl)
:$
(al)
W)
(bl)
(b2)
W)
(3/2)x
(3/2)x
(3/2)x
(3/2)x
Y*
Note that the sum of the fragments of repulsion is 8x+y, which is equal to rHH. To apportion each fragment into the appropriate LAO (8.l)lr0H+ (l-l)/ or LAOS, several general rules are followed: (1) The repulsion between a core orbital, or any orbital exempted from localization (see Selective AO/atom localization), with any LAO on another atom is shared equally between the two LAOS involved. Thus, for the 2x*, one x each goes to hl and i. (2) The equal sharing approach is also practised for LAOS residing on nonbonded atoms. In this case, y* is split evenly between hl and h2. (3) Repulsions arising from nuclear charges assigned to the valence LAOS, with the exception of those exempted from localization (see Selective AO/atom localization), must be shared equally among the valence localized chemical orbitals, valence LCOs. For example, each of 1/l, 112,al, and o2 should receive 3~ from the repulsion between the 1 + charge on each of hl and h2 and the 6+ charge on oxygen. In the case of a bonding LCO, this repulsion is further shared equally between bonding LAO partners. Therefore, 3x assigned to al yields (3/2)x for bl and (3/2)x for hl. The result of this elaborate scheme to obtain EN,aais presented in Table 7. For the free oxygen atom, the ground state ls22s22p4 coincides with the valence state. As a result the promotion energy is zero, dEi = 0, and eqns. (20) and (21) are not applicable. Using the transformation from AOs to LAOS, TA
for Hz0 in eqn. (15), the free oxygen valence state electronic population and energy terms in the LAO basis, qz,, and FAA,&, are derived and shown in Table 7. Using the data in Tables 3 and 7 and eqns. (21)- (26)) the LAO total energy and binding energy terms, Et,ab and Eb,ab,for H,O are obtained and listed in Table 8. The LAO quantities are combined as before to yield the LCO quantities, E,,, and Eb,,d, for Hz0 in Table 9. Summary: the H,O molecules The bonding scheme of two nonequivalent lone pairs for HzO, Qxand Qa (as described above), is interesting but not as “natural” as the scheme of two equivalent lone pairs, 111and 92 (depicted in Fig. 1 (bottom) ). This assessment comes from observing the rather strained angle between the two bonding LAOS, 8(bl,b2), in the former scheme (111.3” ) as compared with the more “natural” angle in the latter scheme (100.7’ ) in consideration of the calculated bond angle (100.0’ ) . A comparison of values in Table 5 for the nonequivalent lone pairs model, against those in Table 4 for the equivalent lone pairs model, reveals many differences. The differences are expected on changing from a roughly sp2 to a roughly sp3 hybridization. For instance, the energy of the bond orbital al, E (al ) , in Table 5 is expected to be lower than the corresponding value in Table 6, owing to a higher 2s orbital character in the al orbital of the former. Certain energy values from the extended A0 basis (4-31G) in Table 6 have become significantly lower than their counterparts from the minimal A0 basis (STO-3G) in Table 4. These include energies of the core and bond orbitals, E(i) and E (al ) . On the other hand, the energy of the lone pair, E (81)) has increased. Again, directions of the observed energy changes are expected for this particular change of A0 basis from STO-3G to 4-31G. With respect to increasing the stability of the bond orbital, al, the split-valence A0 basis has made a definitive contribution. The calculated values based on the localized chemical orbitals, LCOs, obtained from the STO-3G A0 basis are highlighted in Table 10. On the whole, the LAO model has provided some chemically meaningful quantities. The inter-orbital angle between the two lone-pair orbitals (121.5’ ) is substantially greater than the angle between the two bond orbitals (100.7” ) . This is consistent with the valence shell electron pair repulsion (VSEPR) theory in its description of a lone pair being “larger” in size than a bonding pair. Moreover, the angle between the two bond orbitals, 100.7 O,is in good agreement with the calculated bond angle of 100.0’. The population for each lone-pair orbital (2.107) and that for a bond orbital (2.120) confirm the common notion of 2 electrons for a lone pair or an electron-pair bond [ 11. The negative sign of the overlap population between a
230 TABLE 10 The H,O molecule: a summary of the calculated values based on localized chemical orbit& (STO3G)” Interorbital angle, 8 (deg. )
W) (al) (R1,!?2) (Ql,ol) (olJJ2)
121.5 108.2 100.7
Population, q
2.107 2.120 0.0 - 0.076 -0.150
Energy (kcal mol-‘)
E
Et
Eb
- 5254 -4843 -99 140 345
- 4248 - 3392 -99 140 345
- 136 -584 -88 231 405
“The oxygen core orbital, (i) , is not included in this table. Values in this table are taken from text (B),Table4 (qandE),andTableg (E,andE,,).
lone-pair and a bond orbital ( -0.076), and that between two bond orbitals ( - 0.150), are indicative of nonbonding interactions. The energy values, show drastically different magnitudes between one representing the bonded interaction (E( 01) = - 4643 kcal mol- ’ ) and one for the nonbonded interaction (E (al,o2) = 345 kcal mol-’ ). The positive sign for the energy of the nonbonded interaction implies a repulsion. The magnitude of this geminal nonbonded interaction (345 kcal mol-‘) is high compared with the magnitudes employed for conventional thermochemical bond energies, e.g. the 116 kcal mol-’ of Table 1 [2]. The change in magnitudes from electronic (E) to total (Et) and to binding (I&,) energy follows the somewhat arbitrary partitioning technique used in this work. The values for I& are those to be used for a comparison with the thermochemical bond energies. Applying the familiar technique of calculating “gross” quantities [ 41 to electron population and binding energy in Table 10, the following values are obtained: gross q(!Il) =2.03; gross q(a1) =1.97; gross &(S!l) =46 kcal mol-‘; gross &(al) = - 119 kcal mol-‘. The calculated net bonding energy of 119 kcal mol-’ for the O-H bond orbital, has finally become compatible in magnitude with the experimental O-H bond energy of 116 kcal mol-’ for the Hz0 molecule. COMPARISON WITH LMO AND PCILO METHODS
The LAO method has several advantages over its predecessor, the LMO approach [ 12,131. To chemists, the hybrid-like characters of LAOS are conceptually simpler than the multicenter LMOs, especially in depicting the nonbonded interactions. Unlike LMOs, which are confined to closed-shell systems, the LAO method is applicable to open shells. When the size of a molecule increases, the computation of LAOS becomes more efficient than LMOs be-
231
cause the A0 basis on a given atom is fixed regardless of molecular size. (In this case a minimal or extended A0 basis with split valence is assumed.) On the other hand, the LMO method has the practical advantage being independent of the A0 basis. As a consequence, a single algorithm of generating LMOs suffices to treat all types of MOs regardless of the nature of the A0 basis sets. Given the same initial set of AOs, the PCILO methodology [ 71 in its zeroth order description yields less accurate electronic properties than those obtainable from the SCF MOs produced by a variational treatment over the entire A0 set. The LAO method is, in essence, an improved zeroth order description of its counterpart in PCILO because the LAOS are extracted from SCF MOs while leaving the parent SCF MOs intact. With respect to treating multiconfigurational wavefunctions, the PCILO method is definitely superior to the LAO method due to its flexibility in handling electron correlation with the perturbative approach. In summary, the LAO method has many advantages in spite of some obvious limitations. For chemists, it is definitely worthwhile to extract LAOS from a MO wavefunction because these hybrid-like orbitals offer a more lucid interpretation of chemical bonding than do delocalized MOs. Furthermore, the LAO populations and energies provide a direct means to analyze bonded and nonbonded interactions. As shown in this paper, the LAO methodology is sufficiently flexible in its applications and can be expected to bring the necessary quantitative background to many of the current qualitative hypotheses of electronic structure. It is hoped that the extended LAO computer programs [25] will help chemists extract chemically and conceptually useful information from large-scale MO calculations. ACKNOWLEDGMENTS
I am deeply grateful to Professor Keith H. Aufderheide of Oglethorpe University for the initiation and development of the localized atomic orbital theory. I also wish to thank Professor Alan D. Isaacson for his careful readings and valuable comments with regard to the manuscript.
REFERENCES 1 2 3 4 5
G.N. Lewis, J. Am. Chem. Sot., 38 (1916) 762; Valence and the Structure of Atoms and Molecules, The Chemical Catalog Co., New York, 1923. See, e.g., L. Pauling, The Nature of Chemical Bond, Cornell University Press, Ithaca, NY, 1960. C.C.J. Roothaan, Rev. Mod. Phys., 23 (1951) 69. R.S. Mulliken, J. Chem. Phys., 23 (1955) 1833,184l. See, e.g., J.A. Pople and D.L. Beveridge, Approximate Molecular Orbital Theory, McGrawHill, New York, 1970 and references cited therein.
232 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
See, e.g., R.C. Bingham, M.J.S. Dewar, and D.H. Lo, J. Am. Chem. Sot., 97 (1975) 1294, 1285,1302,1307,1311. J.P. Malrieu, in G.A. SegaI (Ed. ) , Semiempirical Methods of Electronic Structure Cakulation, Part A: Techniques, Plenum Press, New York, 1977, pp. 69-103. W.J. Hehre, R.F. Stewart and J.A. Pople, J. Chem. Phys., 51 (1969) 2657. R. Ditchfield, W.J. Hehre and J.A. Pople, J. Chem. Phys., 54 (1971) 724. See, e.g., D.J. DeFrees, B.A. Levi, S.K. PoIlack, W.J. Hehre, J.S. Binkley and J.A. Pople, J. Am. Chem. Sot., 101 (1979) 4085,102 (1980) 2513. See, e.g., 0. Chalvet, R. Daudel, S. Diner and J.P. Mahieu (Eds.), Localization and Delocaiization in Quantum Chemistry, Vol. 1, Reidel, Dordrecht, 1975. K. Ruedenberg, Rev. Mod. Phys., 34 (1962) 326. C. Edmiston and K. Ruedenberg, Rev. Mod. Phys., 35 (1963) 457; J. Chem. Phys., 43 (1965) S97; in P.O. Lowdin (Ed.), Quantum Theory of Atoms, Molecules and Solid State, Academic Press, New York, 1966, p. 263. See, e.g., Philip W. Payne and L.C. Allen, in H.F. Schaefer III (Ed.), Applications of Electronic Structure Theory, Plenum Press, New York, 1977, p. 29, and references cited therein. A. Chung-Phillips, J. Am. Chem. Sot., 101 (1979) 1087. J.W.H. Kao and A. Chung-Phillips, J. Chem. Phys., 63 (1975) 4143,4152. J.W.H. Kao and A. Chung-Phillips, J. Chem. Phys., 65 (1976) 2505. A. Chung-Phillips and J.W.H. Kao, J. Chem. Phys., 71 (1979) 3514. K.H. Aufderheide, J. Chem. Phys., 73 (1980) 1777. K.H. Aufderheide and A. Chung-Phillips, J. Chem. Phys., 73 (1980) 1789; 76 (1982) 1885. K.H. Aufderheide, J. Chem. Phys., 76 (1982) 1897. W.J. Hehre, W.A. Lathan, R. Ditchfield, M.D. Newton, and J.A. Pople, Q.C.P.E., 13 (1973) 236. A. Chung-Phillips and C.J. Eyermann, Q.C.P.E. Bull., 2 (1982) 34. A. Chung-Phillips and K.H. Aufderheide, Q.C.P.E. Bull., 2 (1982) 35. A. Chung-Phillips, unpublished results. M.D. Newton, W.A. Lathan, W.J. Hehre and J.A. Pople, J. Chem. Phys., 52 (1970) 4064. K. Kuchitsu and L.S. Bar-tell, J. Chem. Phys., 36 (1962) 2466. G. Birnbaum and S.K. Chatterjie, J. Appl. Phys., 23 (1952) 220. D.D. Wagman, W.H. Evans, V.B. Parker, I. Halow, W.M. Bailey and R.H. Schuman, Natl. Bur. Stand. (U.S.) Tech. Note 273-3 (1968). T. Shimanouchi, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.), 6 (1967 ), 11 (1967 ), 17 (1968).
211
AN INTERPRETATION OF CHEMICAL BONDING IN A MOLECULAR ORBITAL WAVEFUNCTION*
ALICE CHUNG-PHILLIPS
Department of Chemistry Miami University, Oxford, OH 45056 (U.S.A.) (Received 17 November 1987)
ABSTRACT Localized atomic orbit& (LAOS) are atomic hybrid-like orbital8 derived from self-consistentfield molecular orbitals (MOs); their directional characters offer a lucid interpretation of localized electronic properties. Electronic populations and energies associated with individual LAOS and pairs of LAOS may be combined to yield the corresponding properties for the localized chemical orbitals ( LCOs ) . These properties are particularly useful for the study of bonded and nonbonded interactions within a molecule. In this paper the ab initio LAO methodology is both reviewed and updated. For the purpose of demonstrating different facets of this methodology, details of deriving LAO (and LCO) populations and energies from a minimal A0 basis (STO-3G) are shown for the water (H,O) molecule. Preliminary results from extending the minimal to an extended A0 basis of split-valence (43lG) for H,O are also reported.
INTRODUCTION
The nature of the chemical bond has been indisputably a subject of fundamental interest throughout the history of chemistry. The octet rule postulated by Lewis based on electron-pair bonds [ 1] and the average bond energies deduced by Pauling from thermochemical data [2] are still in popular use. Later developments follow the mainstream of quantum mechanics in the study of electronic structures of molecules. In constructing the electronic wavefunction for a complex molecule, the molecular orbital (MO) method is usually preferred to the valence bond (VB) method. The self-consistent-field (SCF) procedure of deriving MOs as linear combinations of atomic orbitals (LCAOs), standardized by Roothaan [3] in the early 19508, yields quantitative information on the electronic energy of the molecule as a whole and the energies of individual MOs. Subsequent application of the Mulliken population analysis [ 4 J to the LCAO MOs results in electron population terms for each A0 (or atom) and between each pair of AOs *Dedicated to Professor Linus Pauling.
0166-1260/66/$03.50
0 1988 Elsevier Science Publishers B.V.
212
(or atoms) in the molecule. These computed energy and population quantities are very useful in the interpretation of spectroscopic data and photochemical properties of molecules. In regard to the chemists’ concept of chemical bonds [ 1,2] as electron pairs localized in different regions of the molecule, the conventional MO method is inadequate. The problem lies in the fact that MOs are delocalized orbitals which do not convey the concept of directed valence. This problem imposes upon chemists a conceptual barrier in understanding the properties of electron-pair bonds and a practical difficulty in estimating the extent to which nonbonded interactions (those between electron pairs from different localities) determine the relative stabilities of structural or conformational isomers. In the SCF MO LCAO treatment of molecules the typical quantum chemical approach to the study of isomeric species is to use the calculated total energies to establish their relative stabilities and to use populations to provide some measure of general intramolecular interactions. Despite this restrictive mode of approach, great strides have been made in the understanding of molecular stabilities from using the semiempirical complete and intermediate neglect of differential overlap methods (e.g. CNDO and INDO of Pople [ 51, MIND0 of Dewar [ 61, PCILO-CNDO of Malrieu [ 71 and their respective co-workers) and the ab initio MO methods with Gaussian-type basis sets (e.g. STO-3G, 431G, .... and MP2/6-31G* methods of Pople and co-workers [8-lo]. ’ To make the MOs more meaningful in the interpretation of chemical bonds it has been known for some time that the delocalized MOs for a closed-shell configuration can be changed into a set of localized MOs (LMOs) [ll] by means of an orthogonal transformation. The most familiar LMO procedure, that of Edmiston and Ruedenberg [ 12, 131, is based on the minimization of the total interorbital (inter-MO) charge-cloud repulsion in the molecule. This method has been successfully applied within both the ab initio and semiempirical frameworks [ 141. In the past decade the major research efforts of this laboratory have been centered on the study of covalent bonds and their associated properties extracted from molecular orbitals. Initially, a theoretical procedure was formulated to partition the binding energy calculated from a semiempirical MO method into two-atom terms [ 151 for the purpose of finding quantum chemical quantities that can be directly related to the kind of bonded and nonbonded interactions discussed traditionally [ 161. Applications of this procedure to improved CNDO wavefunctions for saturated hydrocarbons, amines, alcohols, and ethers [ 171 produced “bond energies” comparable to those obtained from thermochemical data [ 181. More recently the ab initio method of localized atomic orbitals (LAOS) for atoms in molecules was proposed by Aufderheide [ 191; The LAO method implements the standard SCF MO procedure with an orthogonal transformation of the (orthogonal) A0 basis on an atom to the LAO basis on the same atom.
213
The transformation is based on the maximization of the total interorbital (inter-AO) charge-cloud exchange energy for the given atom in the molecule. The resulting LAOS are hybrid-like core, bonding, and lone-pair orbitals. The LAOS on the same atom are orthogonal, but those on different atoms are nonorthogonal. After the transformation, the original MOs may be expressed directly as linear combinations of LAOS. This enables a natural partition of the calculated electronic population and energy for the molecule into population and energy terms associated with individual LAOS and pairs of LAOS. Applications to small molecules [ 20, 211, using STO-3G wavefunctions, have given strong evidence that the model is well suited to the study of general intramolecular interactions in complex molecules. The LAO algorithm used in connection with STO-3G A0 basis sets from GAUSSIAN 70 [22] has also been made publicly accessible through the Quantum Chemistry Program Exchange [23, 241. In the intervening years the LAO method has been modified and extended in order to develop the model into a practical and versatile theoretical tool for in-depth analyses of molecular electronic structures [ 251. Specifically, the original LAO formulation has been extended from closed-shell saturated molecules to open-shell and conjugated systems for a broader range of applications, and from the minimal A0 basis (STO-3G) to an extended A0 basis of split valence (4-31G) for more accurate studies of energies. Additional efforts have been made in the design of the extended LAO computer program [ 251 for a reduction of computation time through selective A0 (or atom) localization. In this paper, preliminary findings in these areas will be reported. THE ORIGINAL LAO FORMULATION
The criterion for the LAO localization is nonarbitrary; it is a natural extension of the principle of localized orbital correspondence between the LMOs in a molecule [ 12,131 and the LAOS on the free atoms [ 191. The resulting LAOS ‘have been shown to be physically meaningful [20] and transferable among molecules of similar bonding environments [ 211. To facilitate a discussion of the extension of this method, the original formulation will be summarized with conventional notations. Derivation of LAOS
For a molecule with 2n electrons forming a closed-shell configuration, the approximate electronic wavefunction Y may be written as a single Slater determinant over n occupied pairs of spin orbitals, ~~vicx and w&3[ 31 Y(1,2,...2n)=(y/,(l)a(l)W,(2)8(2)...cy,(2n-l)a(2n-l)cy,(2n)B(2n)I (1)
214
In the LCAO scheme the spatial orbitals cyare usually the SCF MOs expanded from a basis set of real AOs $
y=@C
or
Yi=C
cpi
#p
P
(2)
where C contains the linear coefficients. (Indices i, j for MOs and p, V,1, rs for AOs are used.) The electronic energy E may be expressed directly in terms of certain A0 quantities (3) where Pw and HP are the density and core matrix elements, and @~/no) denotes a two-electron interaction integral [ 5 1. The electron distribution in the molecule may be estimated by means of the Mulliken population analysis [ 41. The total number of electrons is decomposed into a sum of terms’
where SW is the overlap integral. The term qp represents the electron charge contributed by the A0 charge density (CD ) , Q&#,,. Suppose now a similar decomposition of the energy E is desired for providing a description of the “energy distribution”. Analogous to eqn. (4)
where Ew must be properly defined so as to represent the energy distribution associated with the A0 CD qJ,&. A close scrutiny of the conventional expression for E as given in eqn. (3), however, reveals that the integral (puo/lv) in the exchange energy term refers to an interaction between A0 CDs #,,@,and &J,, neither of which relates to &,A. To highlight the A0 CD of interest, $,,$,,, the alternative expression for E is used [ 191
Using this new expression the partitioning of E into contributions from individual A0 densities $,&,, as prescribed in eqn. (5) can be realized with
Ew=&vfb + C C (4P&o - i PJ’av ) W/J-a) Note that this new form of expressing the exchange energy offers a means for the localization of AOs as described in eqn. (10) below. The usefulness of eqns. (4) and (5) becomes more evident if, instead of the A0 basis #, a different basis x is used such that x reflects the property of di-
215
rected valence. In this study x is specifically the set of hybrid-like LAOS related to # by an orthogonal transformation T p$T
or
while keeping the MOs invariant ry=@C=xD
and
D=‘i’C
(9)
(Indices a, b, c, d are used for LAOS.) T is a block-diagonal matrix containing one block TA for each atom A in the molecule. 9 is the transpose of T. TA is obtained by an iterative process which maximizes the exchange function LA deduced from eqn. (6). LA evaluated in the A0 basis, #, is I
(10) whereas its maximum value is reached in the LAO basis x (11)
Note that the summation over index ,u or a is over orbitals on atom A only, whereas the summations over indices 2 and o must include all orbitals in the molecule. The resulting TA is orthogonal as the AOs on each atom have been made orthogonal initially. TA defines the composition of the LAOS on atom A in terms of the AOs on the same atom. Using T, the transformation from the A0 basis to the LAO basis for the oneelectron quantity M is carried out as M(X) =T M(#)T
(12)
where M represents the density matrix P, the overlap matrix S, the core-Hamiltonian matrix H, or one of the Cartesian moment matrices X, Y, and 2. As for the two-electron integrals the time-consuming task of full transformation is avoided by using, for example, the mixed-basis quantities P,, Pb, and (au/ Aa) as in eqn. (11) which, upon multiplication and then summation over all A0 charge densities @J&,,become equivalent to using pure LAO quantities Pa,, Pbaand (aa/bc) over all LAO charge densities xbxc. The diagonal elements of X(x), Y (2) , and Z (x) are next used to determine the direction and mean distance r(a) of an electron in the LAO xa relative to the atomic nucleus. From these directions, the angle between two LAOS X~and xb on the same atom, 0( a$), can be deduced. These terms provide a geometrical description of the LAOS.
216
LAO (or LCO) electronic populations and energies
To study localized properties, a knowledge of electronic population and energy distributions over atomic-core, lone-pair, bonding, and other chemically significant regions is essential. Thus, it is important to partition the total electronic charge, 2n, and the electronic energy, E, in the LAO basis
(134 where qab = Pab Sab
(13b) (14a)
(14b) Transformations described in, and subsequent to, eqn. (12) are used in the calculation of the one- and two-electron quantities in the above two equations. The LAO population and energy terms (q& and Eab, respectively) furnish a convenient set of building blocks for constructing interatomic bonding terms by simple summations. Proper groupings of certain valence LAOS on different atoms will not only illuminate the bonding features but also reduce the number of significant terms for subsequent analyses. These groupings, plus the core and lone-pair LAOS, may be called the localized chemical orbitals (LCOs) . (Indices y, 6 are used for LCO terms.) Example: the Hz0 molecule
Presently it is instructive to use a numerical example for clarifying notations and the major steps in the LAO procedure. For this purpose the water (H,O) molecule is selected owing to its simplicity and importance in chemistry. Furthermore, a wavefunction expanded from a minimal A0 basis, STO-3G, is chosen due to its availability [ 8,221. It should be emphasized that the numerical results from this exercise will not be sufficiently accurate because of the nature of the wavefunction. If the wavefunction has taken into account electron correlation [lo]and is also constructed from an extended A0 basis including polarization functions, then the results can be expected to approach chemical accuracy. Hz0 has the experimental geometry of 0.957 A for the bond length and 104.5’ for the bond angle, which may be compared to the STO-3G optimum geometry
217
of 0.990 A and 100.0” [8]. Experimental and STO-3G data pertinent to H,O are tabulated in Tables 1 and 2. In Hz0 there are seven AOs: five on the oxygen atom (Is, 2s, 2px, 2py, 2pz), and one on each hydrogen atom (Is). (Note that the 1s and 2s orbitals on oxygen have been made mutually orthogonal subsequent to the STO-3G calculation.) The valence AOs include the oxygen 2s, 2px, 2py, and 2p.z orbitals plus the two hydrogen 1s orbitals designated as hl and h2. These six valence AOs are schematically represented in Fig. 1 (top). From the SCF MOs, 4 valence LAOS for oxygen are derived from the 4 valence AOs with the frozen-atomic-cores approximation (vide infra) via eqns. ( 10) and ( 11). These are the lone-pair LAOS, Pl and 1/2,and the bonding LAOS, bl and b2 a1 =a(2s) -b(2px)
+c(2pz)
Q2=a(2s) -b(2px) -c(2pz) bl=a(2s) +b(2px) +c(2py) b2=a(2s)
(15)
+b(2px) -c(2py)
where a= 0.58565, b = 0.39624 and c = 0.70711. The associated average electron distances and interorbital angles related to the LAOS are: r(Q1) = 0.325 A, r(bl)=0.249 A, 13($?l,Q2)=121.5”, kQl,bl)=108.2”, and 8(bl,b2)=100.7”. The remaining AOs are: i, representing the oxygen 1s core orbital; and hl and h2, the hydrogen 1s orbitals. These three AOs become LAOS automatically since they require no localization. The valence LAOS are displayed in Fig. 1 (center). The partitioned charges, qab,and energies, Ea6, for the 7 LAOS (i, Ql, 112,bl, b2, hl, h2) based on eqns. (13) and (14) are presented in Table 3. Note that the tabulated values Ma,,, where M=q or E, are expressed as M(a) = Ma, for TABLE 1 The Hz0 molecule: STO-3G values versus experimental values8*b Quantity
Experimental value
STO-3G value
Geometry t(O-H) (A) B(HOH) (deg.) Dipole moment (D ) Atomization enen@ (kcal mol- ’ ) O-H bond energyd (kcal mol-’ )
0.957 104.5 1.846 232.2 116
0.990 100.0 1.51 143.4 72
“STO-3G values for the optimum geometry were taken from ref. 26. Other values are calculated for this work based on the STO-3G optimum geometry. bExperimental values were taken from: for geometry, ref. 27; for dipole moment, ref. 28; for atomization energy, refs. 29,30. “Energy for the reaction, H,O (g) +2H (g) +0(g) at 0 K and corrected for zero point energy. %&en as onehalf of the atomization energy.
218 TABLE 2 Pertinent data (a.u.) from STO-3G calculations for H,O A. Optimum geometry (in Cartesian coordinates) Y 0 f.6 0.0 f.0 Hl 1.20257 1.43317 0.0 H2 1.20257 - 1.43317 0.0 &ZB
B. Nuclear repulsion: 5 g 1;\g=8.96107 ‘OZH ‘*’ -=1.87068 rOH
~427610 *
z,zH=~=O_3488f$ rHH
2.86633
C. Free atom energies: F Ei= - 74.73731 0: E= - 73.86415 H: E = -0.46658 D. Electronic energy: E = - 83.86698 Et = E + (nuclear repulsion) Total energy: = - 74.96591 Binding energy: Eb = Et- (free atom energies) = -0.22860
Fig. 1. Schematic representations of the valence orbitals of H,O: top, the atomic orbit& (AO); center, the localized atomic orbit& (LAOS); bottom, the localized chemical orbit& (LCOs ) .
219 TABLE 3 The H,O molecule: LAO electronic population and energy terms (STG-3G)B*b (i)
(Ql)
(bl)
(hl)
- 52.945 2.005
- 2.107 8.374
- 0.802 3.250
- 0.627 1.731
(i,Ql)
(W2)
(bl,b2)
(hl,h2)
-0.158 0.0
-0.052 0.0
-0.091 0.333
(Pl,bl)
(blhl)
- 0.0 0.089
- 0.692 2.737
(Chl)
(blh2)
-0.076 0.312
- 0.030 0.134
- 0.082 0.0 (i&l) - 0.0 0.022 (i&l) - 0.004 0.092
“Energies are given in a.u. Notations are defined in text. Terms equivalent by spatial symmetry to those listed are not shown. The same convention is followed for Tables 4-10. bDefinitions:N=~q(a)+~~q(a,b)andE=~E(a)+~~E(a,b)whereNisthetotalnumber of electrons an: E is the:konic energy. Vkes for tii*ppulation terms, q(u) and q(a,b), and energy terms, E(a) and E(a,b) , based on the localized atomic orbit.& L, xb, am listed under theLAOlabels (a) and (a,b).NotethatforM=qorE,M(~)=M,andM(a,b)=M~~+&,in relation to eqns. (13) and (14).
TABLE 4 The Hz0 molecule: LCO electronic population and energy terms ( STO-3G)“*b
ti)
(al)
(01)
:
- 52.945 2.005
2.107 - 8.374
2.120 - 7.718
:
(i,Ql) -0.082 0.0
(QW) 0.0 -0.158
(ol@) -0.150 0.550
(i,ol)
(QlPl) - 0.076 0.223
z
- 0.004 0.070
“See footnote a of Table 3. bDefinitions: N= C q(y) + TcT q (y,6) and E = C E( y) + T<& E( y,6) Y Y where N is the total number of electron and E is the electronic energy. Values for the population terms, q(y) and q(yJ), and energy terms, E(y) and E(yJ), based on the localized chemical orbitals, are listed under the LCO labels (y) and (~,a). Note that for ME q or E, M(y) = M, and MY,& =A&+ i+&
a= b and M(a,b) = M,,b + Mb,, for a # b. The terms equivalent by spatial symmetry to those shown (for (Q2), (b2), .... (b2,hl) ) have been omitted from the list. Summation of the 28 LAO terms (7 X 8/2) gives 10 electrons for the charge and -83.867 au. for the energy. Grouping each bonding LAO on oxygen with the hydrogen Is orbital in its directed path leads to a “o-bond orbital”. Symbolically, the two bond orbitals are: al = bl + hl and a2 = b2 + h2, where + defines how a calculated M(a) or M(a,b) should be summed. Thus, the population in al is: q(a1) =q(bl) +q(bl,hl) +q(hl). Likewise, the overlap population between 01 and a2 is: q(al,o2) =q(bl,b2) +q(bl,h2) +q(b2,hl) +q(hl,h2). Applying this procedure, the 4 LAOS (bl, b2, hl, h2) are, in essence, replaced by 2 bondorbitals (al, o2), making a total of 5 LCOs (i, Ql, Q2, al, 02). The 4 valence LCOs (al, Q2, 01, o2) are symbolically represented in Fig. 1 (bottom). The LCO population and energy terms, qysand I&, are given in Table 4. Note that the LCO terms for (i), (al), (i,Ql), and (Ql,Q2) are not different from the previous LAO terms. There are a total of 15 LCO terms (5 x 6/2), a reduction of nearly one half compared to the number of the LAO terms. The calculated values for Hz0 as exhibited here are illustrative of the kind of numbers sought by quantum chemists for quantifying popular chemical concepts and trends. For instance, the average distance of an electron in a lonepair orbital on oxygen (r (Ql) or F (as) ) and their relative orientation [ 8( Ql,Q2)] influence the geometry of hydrogen-bonded clusters in’ an aqueous solution. The charges and energies in bonded regions [ (al ), (02) ] and nonbonded regions ((Ql,ol), (Ill,&!), (Q2,01), (112&Z),(al,o2)),versus those relatedto the lone pairs [ (Ql), (Q2), (Ql,Q2)1, affect the O-H bond strength and the geometry of H,O. EXTENSIONS OF THE LAO METHOD
To make the LAO method a practical and versatile theoretical tool, many fundamental problems need to be resolved. In the following, preliminary reports are given for advances made in certain areas: applications for chemical systems other than the closed-shell saturated compounds (first and second sections), reduction of computing time to allow for treating molecules with large numbers of electrons (second section), extending the A0 basis to a splitvalence basis for better accuracy (third section ), and the partitioning of total and binding energies for studies of bonded and nonbonded interactions (fourth section). Open-shell configuration
For an open shell there are two separate density matrices, P (@)”and P (qb)p for the cy and /3 electrons, respectively. The full density matrix, P (#), is the sum of these two
221
To derive the LAOS we propose to carry out the maximization of LA using P (@) in eqn. (10) and arrive at a single T which is to serve both cy and p electrons. For subsequent calculations of LAO quantities we will apply T to P(@)“, P(@)? P(#), and other A0 quantities. Finally, the qaband Eabterms are obtained from eqns. (13) and (14) except for the replacement of (l/ 4)P,,PAb in the exchange energy term by ( 1/8)Pa0LYPAb(Y+ ( 1/8)P,BPA68. This open-shell procedure has already been incorporated in the LAO computer program [24]. Selective AOlatom localization
The AOs on atom A may be separated into subsets (p), (v), etc. One may choose to localize a particular subset (p). To do so, one simply excludes all AOs on A not in (p) in the first summation for LA(#) in eqn. (lo), but keeps the other two summations (over 1 and o) intact (The latter summations over all A0 charge densities #,&, accounts for the full potential field of the molecule as determined by the SCF MOs.) One form of this approach is the approximation of frozen atomic cores (FACs), which is an atomic core-valence separation based on energy. The core AOs are assumed to be localized sufficiently to be taken as the core LAOS at the outset. This approximation effects considerable savings in computational efforts and also produces sufficiently accurate results. For Hz0 (vide supra: Example: the H,O molecule), a six-fold time reduction is achieved through FACs with a difference of 0.005 electron in q(!?l) or q(b1) from the complete treatment [ 201. The FACs treatment has been well tested [ 241. Another familiar example of this approach (based on symmetry) is the 0-7~ separation in a conjugated molecule, where the n-AOs are equated to the JZLAOS at the outset, leaving the o-AOs to be localized. For each carbon atom in benzene (C&H,) for example, the 2pz A0 perpendicular to the molecular plane is treated as a K-LAO, while the 2s, 2px, and 2py AOs are localized to yield the three bonding O-LAOS. Analogously, the 2pz A0 for a Hz0 molecule lying in the xy plane may be made into a lone-pair LAO at the outset, yielding a description of two nonequivalent lone pairs in H,O. To test this bonding scheme for H20, the STO-3G wavefunction (see, Example: the H20 molecule) is subjected to this selective A0 localization; i.e. excluding (or freezing) the 2pz A0 from localization. In other words, 2pz is made into the lone-pair LAO, ll~, at the outset. The transformation for the 3 valence AOs on oxygen to the new
222 TABLE 5 The Hz0 molecule: LCO electronic population and energy terms with selected A0 localization (STO-3G)’
(i)
(Qlr)
(Qu)
(al)
:
- 52.945 2.005
2.000 -7.568
2.097 - 8.327
2.154 - 7.954
(i,Qn) -0.025 0.0
(Qn,Qa) 0.0 -0.112
(QWl )
:
-0.051 0.155
(&a2) -0.099 0.372
:
(i&Q -0.016 0.0
(Qwl) 0.0 -0.059
:
(Cal ) --0.031 0.002
“See footnote a of Table 3 and definitions (footnote b ) in Table 4.
lone-pair LAO, Ilo, and the bonding LAOS, bl and b2 is given as Qa=a’ (2s) +b’ (2p3c) bl=a(2s)+b(2px)
+c(2py)
b2=a(2s)+b(2px)
-c(2py)
(16)
where a’ = 0.66328, b’ = -0.73016, a=0.51630, b=0.48315 and c=O.70711. The associated average electron distances and interorbital angles related to the new LAOS are: r(Qa)=0.342 A, r(bl)=0.303 A, 8(Qa,bl)=124.3”, and 8(bl,b2) = 111.3”. The LCO population and energy terms based on this set of LAOS are shown in Table 5. The atoms in a molecule may be grouped according to symmetry. In each group, one may select only one atom for maximization of LA and then relate the resulting LAOS to all other symmetry-equivalent atoms via orthogonal transformations. For example, only one carbon atom in benzene needs to be treated explicitly. In fact, a complete freedom exists in the choice of atoms for localization regardless of symmetry. In pyridine, if the “reactivity” of the lone pair on nitrogen is the sole interest, the carbon atoms will not require explicit LAO considerations. This option is particularly convenient and economical for certain comparative studies in related molecules. Extended A0 basis The single-configuration wavefunction spanned by a minimal A0 basis, as employed in the current LAO method, is inadequate for the determination of very accurate relative energies. It is well known that the use of configuration
223
interaction (CI) or some variant, coupled with an extended A0 basis, will improve the wavefunction in this respect. These added features to the original wavefunction would mean the following in deriving the LAOS. The upper limits for all summations over the A0 or LAO indices in the prior equations must be numerically increased. The density matrix elements must now reflect contributions from more than one configuration or weighted occupations of MOs. The cost in computation for these additional features, if drastic measures are not taken to modify the iterative step represented by eqns. (10) and (ll), would render the LAO procedure totally impractical for large systems. The specific LAO treatment of a CI wavefunction and/or the use of polarization functions in the extended basis will be postponed. At present, the interest in practical applications calls for an extended A0 basis containing splitvalence, or double-zeta, orbitals at the level of 4-31G [ 91. A “short-cut” approach which gives reasonable results is described below. Take the HZ0 molecule (vide supra, Example: the Hz0 molecule), again, as an example. For the 4-3lG basis there are 8 valence AOs for oxygen, falling into two sets: (2sI,2pxI, 2pyI,2pzI) of Set I and (2sO,2p~O, 2pyO,2pzO) of Set 0, corresponding to replacing each orbital exponent (zeta) in the minimal A0 basis by a larger (Set I) and a smaller (Set 0) exponent. To obtain preliminary LAOS for the 4-31G calculation for HZ0 (based on the STO-3G optimum geometry as in the Example section), the A0 coefficients for the LAOS from the STO-3G calculation in eqn. (15) may be directly transferred to Set I and Set 0 to yield two sets of 4 LAOS: (QlI, Q21,bl1, b21) of Set I and ($90, P20, b10, b20) of Set 0. To obtain LAO charge and energy terms, the valence LAOS in Set I and Set 0 on oxygen are merged into a single set of “four” orbitals based on transformation properties: {QlI, alO>, (P21, !?20}, (bl1, blO}, and (b21, b20). Adding to these four orbitals the core orbital of oxygen, {i}, and the two hydrogen orbitals, (hl1, h10) and (h21, h20), “seven” LAOS are again obtained as in the case of a minimal A0 basis. In this extended basis, each calculated LAO term contains contributions from components I and 0. Results for the LCO analysis based on the 4-31G basis with this procedure are presented in Table 6. If greater accuracy is desired, the localization procedure may be applied selectively on AOs of the same angular characteristics first, e.g. between 2sI and 2sOto get 2s’ and 2s II, between 2~x1 and 2~x0 to get 2px’ and 2px”, etc. Next, use the A0 coefficients for some “standard” LAOS (e.g. those obtained from STO-3G calculations in eqn. (15) ) as initial input for the iterative LAO derivation on the set with lower energies (2s’) 2px’, 2py’, 2~2’ ) . Finally, apply the transformation matrix thus obtained for the previous set directly to the set of higher energies (2s II,2px”, 2py”, 2~2” ) without going through further iterative LAO derivation. In short, different procedures are available for treating extended A0 sets.
224 TABLE 6 The Hz0 molecule: LCO electronic population and energy terms using an extended A0 basis of split-valence (4-3lG)’
6)
(al)
(al)
:
- 53.295 1.997
2.111 - 8.246
2.096 -7.911
:
(i,Ql) - 0.087 0.0
(QW) 0.0 -0.138
(ol@) -0.136 0.498
;
(i,al) - 0.051 0.0
(QlPl) -0.069 0.180
‘See footnote a of Table 3 and definitions (footnote b ) in Table 4.
The complete treatment, i.e. using all the AOs of a 4-3lG basis in the iterative LAO derivation at once, is unjustified in view of its unreasonable demand on computer resources. The most expedient way is that which has been shown for the H,O molecule. Partitioning of total and binding energies
In order to model energy variations arising from isomeric changes, energy quantities more illustrative than the electronic energy E should be considered. These are the total energy Et (E plus the nuclear repulsion) and the binding energy E,, (Et minus the free-atom energies) in a.u. E,=E+C
&F
(17a)
and &=E,--C
E;;
(17b)
A
(Indices A, B are used for atoms. ) To assess regional contributions to the energy variations, Et and Eb should also be partitioned in the LAO basis yielding &,ab and &,ob analogous to E and Eab in eqn. (14). To do so, unfortunately, calls for arbitrary assumptions with regard to the partitioning of the nuclear repulsion and free-atom energy terms in eqn. (17). For initial trials, the following approaches may be adopted. The free-atom energy in its ground state, E& may be replaced by PA, the energy of the valence state as determined by the bonding environment of A in the molecule, and LIE;, the promotion energy
225
Eb=Et-
c A
(PA-AEi)
(18)
The calculation of E*Ais next carried out for the best estimated valence state. (As an example, for C in CHI, E; corresponds to the ls22s22p2 3P state and E*A,the ls22s2p3 4S state). In the LAO context the valence state is defined by the transformation TA derived for the molecule in question. Accordingly, the charge density obtained for the free-atom valence state, PX, is transformed from the A0 basis to the LAO basis as in eqn. (12) PX(X)=*APZ(#)TA
(19)
Other A0 quantities have already been similarly transformed for the molecular calculation in eqn. (12). Using these transformed quantities the LAO charge and energy terms, qz,aa and E*A& are obtained via eqns. (13) and (14). (Note that q&, = 0 for a # b because S&, = 0. ) In the absence of a better scheme at present, the partitioning of the promotion energy follows a more ad hoc route. A simple equation containing a weighing factor w, may be tested first [ 151 AE+CA
w, AEX
For simplicity, consider the case of FACs (vide supra). Then, for the core LAO (eqn. 21a) and the valence LAO (eqn. 21b), respectively w,= 0
(21a)
and w,=
&z, number of valence electrons on A
(21b)
for which C Aq&a = number of valence electrons on A a over the valence LAO index a for atom A. This approach is based on the assumption that a valence LAO carrying a greater charge has received a greater share of the promotion energy. (For C in CH4, w,= l/4 for each valence LAO.) There is no simple prescription for resolving the nuclear repulsion term, One approach is to split up the nuclear charge on each atom into zAzB/rAB* different LAOS; each LAO receives a certain amount depending on the nature of the LAO. Thus, for ZA on atom A and .& on atom B zA=xAza a
and
(22a)
226
(22b)
ZB=p
(For C in CH4, Z,= 2 for the core orbital and Z,= 1 for each of the bonding LAO, whereas for H, Zb= 1 for the 1s orbital.) The final partition must be in the form
e= c*
F B‘%,bb
EN,~, +
a
implying that the nuclear repulsion will affect only the “diagonal” electronic E, and Ebb, but not the “Off-diagOnai” tk?rIIM, &b and Eba. Specific details for EN,m and EN,bbneed to be worked out for each group of related molecules under study, e.g. molecules in the series CnHzn+z, AH, or H,ABH,, where A, B = C, 0, N, F, etc. To summarize, the partitioning in the LAO basis for Et and E,, energieS,
and
Eb= 5 F %,a6 involves the following approximations: E t,ob
=
‘%b
+
6%.ab
for a and b both on A @Sal
)hb
and E b,ab =Et,ob
-
[E*A,ab-
(W%ik&b
1
(25b)
andforaonAandbonB E t,ab =E&
(264
and E b,ab =
Et,ab
(26b)
The justification for any of these approximations must come from a high degree of success in explaining the physical origins of the observed energy variations in isomeric species [ 141.It is foreseeable that a reasonable scheme for the energy partitioning will materialize from future trial calculations. As a numerical example, the H,O molecule (vide supra) is again used for further demonstration. The nuclear repulsion is handled in a way that will allow the binding energy of a bond orbital 01, Eb (cd), to be lower than that of a lone-pair orbital Ill, Eb (II1) . The details are given next. The 8+ charge on oxygen is partitioned as follows: 2 + for the core LAO and 3/2 + for each of the four valence LAOS. For hydrogen, 1 + is assigned to the
227 TABLE 7 Pertinent data for use in partitioning A. Nuclear repulsiond (i) 2x EN 1.069 B. Free oxygen atom (i) 2.000 : - 51.787
the total and binding energies of Hz0 (STO-3G)“a*c*d
(al)
(bl) (3/2)x
3x 1.604
WI (5/2)~+(1/2)~ 1.511
0.802
(Ql)
(bl)
1.843 - 6.553
1.157 -4.002 (bLb2) 0.097
E
(i,Ql) -0.068
(w!2) -0.017
E
($1) -0.039
(Qlbl) -0.145
‘See Table 3 for references and the text for details. YI’he nuclear repulsion (-8.90107 a.u.) and electronic energy of oxygen ( -‘73.89415 a.u.) given in Table 2 are partitioned into LAO energy terms. The LAO population terms for oxygen are listed for the nonxero terms only. “For the free hydrogenatom, q(h1) =l.OOOand E(h1) = -0.467 a.u. dFrom Table 2 B: z=l/ron=0.53452a.u. andy= l/ruu=0.34888a.u.
TABLE 8 The H,O molecule: LAO total and binding energy terms (ST@3G)”
Et & Et Eb
6)
(al)
(bl)
(hl)
-51.876 - 0.089
-6.770
-2.448
-0.220
W 1
(QW 1
(bl&‘)
-0.158
-0.014
-0.141
-0.051 - 0.045
-0.022 0.016
(i,hl) Et Eb
1.554
-0.082
fib1 1 Et Eb
-0.217
0.092 0.092
U&b11 - 0.089 0.056
(Ql,hl)
(bl,hl) -2.737 -2.737
(bl,hZ)
0.312
0.134
0.312
0.134
“See Table 3 for references and the text for details.
0.247
W&2) 0.333 0.333
228 TABLE 9 The H,O molecule: LCO total and binding energy terms (STO-3G)’
6) Et &
- 53.876 - 0.089
(al)
-6.770 -0.217
(01)
Wl)
W,L2)
(al@)
-0.082 -0.014
Et &I
-0.158 -0.141
(i,ol) E!J
0.550 0.646
W,al)
0.070 0.109
Et
- 5.406 - 0.936
0.223 0.368
*See Table 4 for references and the text for details.
hl or h2 orbital with no ambiguity. Let X= l/roH and y= l/~r..m, in a.u.. The repulsions between 1+ charge in hl with the 8+ charge in the oxygen LAOS and with the 1+ charge in h2 can be broken down in the LAO representation as (hl)
:$
(al)
W)
(bl)
(b2)
W)
(3/2)x
(3/2)x
(3/2)x
(3/2)x
Y*
Note that the sum of the fragments of repulsion is 8x+y, which is equal to rHH. To apportion each fragment into the appropriate LAO (8.l)lr0H+ (l-l)/ or LAOS, several general rules are followed: (1) The repulsion between a core orbital, or any orbital exempted from localization (see Selective AO/atom localization), with any LAO on another atom is shared equally between the two LAOS involved. Thus, for the 2x*, one x each goes to hl and i. (2) The equal sharing approach is also practised for LAOS residing on nonbonded atoms. In this case, y* is split evenly between hl and h2. (3) Repulsions arising from nuclear charges assigned to the valence LAOS, with the exception of those exempted from localization (see Selective AO/atom localization), must be shared equally among the valence localized chemical orbitals, valence LCOs. For example, each of 1/l, 112,al, and o2 should receive 3~ from the repulsion between the 1 + charge on each of hl and h2 and the 6+ charge on oxygen. In the case of a bonding LCO, this repulsion is further shared equally between bonding LAO partners. Therefore, 3x assigned to al yields (3/2)x for bl and (3/2)x for hl. The result of this elaborate scheme to obtain EN,aais presented in Table 7. For the free oxygen atom, the ground state ls22s22p4 coincides with the valence state. As a result the promotion energy is zero, dEi = 0, and eqns. (20) and (21) are not applicable. Using the transformation from AOs to LAOS, TA
for Hz0 in eqn. (15), the free oxygen valence state electronic population and energy terms in the LAO basis, qz,, and FAA,&, are derived and shown in Table 7. Using the data in Tables 3 and 7 and eqns. (21)- (26)) the LAO total energy and binding energy terms, Et,ab and Eb,ab,for H,O are obtained and listed in Table 8. The LAO quantities are combined as before to yield the LCO quantities, E,,, and Eb,,d, for Hz0 in Table 9. Summary: the H,O molecules The bonding scheme of two nonequivalent lone pairs for HzO, Qxand Qa (as described above), is interesting but not as “natural” as the scheme of two equivalent lone pairs, 111and 92 (depicted in Fig. 1 (bottom) ). This assessment comes from observing the rather strained angle between the two bonding LAOS, 8(bl,b2), in the former scheme (111.3” ) as compared with the more “natural” angle in the latter scheme (100.7’ ) in consideration of the calculated bond angle (100.0’ ) . A comparison of values in Table 5 for the nonequivalent lone pairs model, against those in Table 4 for the equivalent lone pairs model, reveals many differences. The differences are expected on changing from a roughly sp2 to a roughly sp3 hybridization. For instance, the energy of the bond orbital al, E (al ) , in Table 5 is expected to be lower than the corresponding value in Table 6, owing to a higher 2s orbital character in the al orbital of the former. Certain energy values from the extended A0 basis (4-31G) in Table 6 have become significantly lower than their counterparts from the minimal A0 basis (STO-3G) in Table 4. These include energies of the core and bond orbitals, E(i) and E (al ) . On the other hand, the energy of the lone pair, E (81)) has increased. Again, directions of the observed energy changes are expected for this particular change of A0 basis from STO-3G to 4-31G. With respect to increasing the stability of the bond orbital, al, the split-valence A0 basis has made a definitive contribution. The calculated values based on the localized chemical orbitals, LCOs, obtained from the STO-3G A0 basis are highlighted in Table 10. On the whole, the LAO model has provided some chemically meaningful quantities. The inter-orbital angle between the two lone-pair orbitals (121.5’ ) is substantially greater than the angle between the two bond orbitals (100.7” ) . This is consistent with the valence shell electron pair repulsion (VSEPR) theory in its description of a lone pair being “larger” in size than a bonding pair. Moreover, the angle between the two bond orbitals, 100.7 O,is in good agreement with the calculated bond angle of 100.0’. The population for each lone-pair orbital (2.107) and that for a bond orbital (2.120) confirm the common notion of 2 electrons for a lone pair or an electron-pair bond [ 11. The negative sign of the overlap population between a
230 TABLE 10 The H,O molecule: a summary of the calculated values based on localized chemical orbit& (STO3G)” Interorbital angle, 8 (deg. )
W) (al) (R1,!?2) (Ql,ol) (olJJ2)
121.5 108.2 100.7
Population, q
2.107 2.120 0.0 - 0.076 -0.150
Energy (kcal mol-‘)
E
Et
Eb
- 5254 -4843 -99 140 345
- 4248 - 3392 -99 140 345
- 136 -584 -88 231 405
“The oxygen core orbital, (i) , is not included in this table. Values in this table are taken from text (B),Table4 (qandE),andTableg (E,andE,,).
lone-pair and a bond orbital ( -0.076), and that between two bond orbitals ( - 0.150), are indicative of nonbonding interactions. The energy values, show drastically different magnitudes between one representing the bonded interaction (E( 01) = - 4643 kcal mol- ’ ) and one for the nonbonded interaction (E (al,o2) = 345 kcal mol-’ ). The positive sign for the energy of the nonbonded interaction implies a repulsion. The magnitude of this geminal nonbonded interaction (345 kcal mol-‘) is high compared with the magnitudes employed for conventional thermochemical bond energies, e.g. the 116 kcal mol-’ of Table 1 [2]. The change in magnitudes from electronic (E) to total (Et) and to binding (I&,) energy follows the somewhat arbitrary partitioning technique used in this work. The values for I& are those to be used for a comparison with the thermochemical bond energies. Applying the familiar technique of calculating “gross” quantities [ 41 to electron population and binding energy in Table 10, the following values are obtained: gross q(!Il) =2.03; gross q(a1) =1.97; gross &(S!l) =46 kcal mol-‘; gross &(al) = - 119 kcal mol-‘. The calculated net bonding energy of 119 kcal mol-’ for the O-H bond orbital, has finally become compatible in magnitude with the experimental O-H bond energy of 116 kcal mol-’ for the Hz0 molecule. COMPARISON WITH LMO AND PCILO METHODS
The LAO method has several advantages over its predecessor, the LMO approach [ 12,131. To chemists, the hybrid-like characters of LAOS are conceptually simpler than the multicenter LMOs, especially in depicting the nonbonded interactions. Unlike LMOs, which are confined to closed-shell systems, the LAO method is applicable to open shells. When the size of a molecule increases, the computation of LAOS becomes more efficient than LMOs be-
231
cause the A0 basis on a given atom is fixed regardless of molecular size. (In this case a minimal or extended A0 basis with split valence is assumed.) On the other hand, the LMO method has the practical advantage being independent of the A0 basis. As a consequence, a single algorithm of generating LMOs suffices to treat all types of MOs regardless of the nature of the A0 basis sets. Given the same initial set of AOs, the PCILO methodology [ 71 in its zeroth order description yields less accurate electronic properties than those obtainable from the SCF MOs produced by a variational treatment over the entire A0 set. The LAO method is, in essence, an improved zeroth order description of its counterpart in PCILO because the LAOS are extracted from SCF MOs while leaving the parent SCF MOs intact. With respect to treating multiconfigurational wavefunctions, the PCILO method is definitely superior to the LAO method due to its flexibility in handling electron correlation with the perturbative approach. In summary, the LAO method has many advantages in spite of some obvious limitations. For chemists, it is definitely worthwhile to extract LAOS from a MO wavefunction because these hybrid-like orbitals offer a more lucid interpretation of chemical bonding than do delocalized MOs. Furthermore, the LAO populations and energies provide a direct means to analyze bonded and nonbonded interactions. As shown in this paper, the LAO methodology is sufficiently flexible in its applications and can be expected to bring the necessary quantitative background to many of the current qualitative hypotheses of electronic structure. It is hoped that the extended LAO computer programs [25] will help chemists extract chemically and conceptually useful information from large-scale MO calculations. ACKNOWLEDGMENTS
I am deeply grateful to Professor Keith H. Aufderheide of Oglethorpe University for the initiation and development of the localized atomic orbital theory. I also wish to thank Professor Alan D. Isaacson for his careful readings and valuable comments with regard to the manuscript.
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G.N. Lewis, J. Am. Chem. Sot., 38 (1916) 762; Valence and the Structure of Atoms and Molecules, The Chemical Catalog Co., New York, 1923. See, e.g., L. Pauling, The Nature of Chemical Bond, Cornell University Press, Ithaca, NY, 1960. C.C.J. Roothaan, Rev. Mod. Phys., 23 (1951) 69. R.S. Mulliken, J. Chem. Phys., 23 (1955) 1833,184l. See, e.g., J.A. Pople and D.L. Beveridge, Approximate Molecular Orbital Theory, McGrawHill, New York, 1970 and references cited therein.
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