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Approximate molecular orbital theory: the ese MO formalism. I. General theory and notation
CHEMICAL
PHYSICS
4 (1974)
220-2X.0
NORTH-HOLLAND
PUBLISHING
COMPANY
APPROXIMATE MOLECULAR ORBITAL THEORY: THE ESE MO FORMALISM. I. GENERAL THEORY AND NOTATION P.G. BURTON Chemisrry
Department
t. Wollongong University
College, Wollongong, N.S. W. 2500, Australia
and R.D. BROWN Chemistry
Department.
hfonash University,
C&ytorr,
Victoria
3168.
Australta
Received 29 November 1973 Revised manuscript received 25 February 1974
A new formalism for simplified molecular orbital calculations is elucibted. ‘Ihe focus of the formalism is the of a good approximation to the LCAO SCF F matrix of Roothaan’s equations, and preoccupation with approximations to individual molecubr intepals is avoided. The geat majority of multicentre two-electron integrals of the exact formalism are found to be largely inconsequential to the attainment of a good approsimation to ihe F production
matrix.
I. Introduction Relatively littlc is known of the precise role of different classes of the multicentre integrals that occur in large numbers in the widely used LCAO SCF h10 appl&mation [I ] to the Hartree-Fock (HF) theory of molecular electronic structure calculation. The accuracy and reliability of any LCAO SCF MO calculation (relative to the “HF limit”) critically depends on the nature, extent and flexibility of the chosen basis set, in terms of which the molecular orbitals are expanded. However, as the size of the basis set is increased IO approach the HF limit, it is well known that the number of integrals over the expansion basis functions situated at different points in the molecule that must be evaluated, stored and manipulated increases dramatically. Many simplified LCAO SCF MO procedures have evolved which seek to reduce the complexity of an exact (ab initio) solution of the LCAO SCF MO method, by ignoring the contributions of some intet Permanent address to which all correspondence directed.
should be
grals that are either difficult to evaluate or that occur in inconveniently large numbers in the exact caiculation of the LCAO SCF wavefunction. The most widely used such methods have been based on the so-called CNDO and lND0 integral approximations, which together with the more complex NDDO approximation, constitute the general group of “neglect of differential overlap” (NDO) approximations [2]. Brown and Burton [3] have recently noted an analysis of the role that various classes of muiticentre two electron integrals play in the generation of the final SCF wavefunction. That analysis, based on simple physical and arithmetic arguments, casts strong doubt on the reliability and predictive capability of all approximate MO methods based on ND0 approximations. The basic premise of the analysis was that it is the shucfufe of the LCAO SCF F matrix (the size of the individual F matrix elements, their sign, and especially the rebfionship between different - particularly diagonal versus off-diagonal - F matrix elements) which determines the ultimate generation of covalency in the final wavefunction. Accordingly the aim of any simplified or approximate molecular orbital method must always be to produce at SCF convergence an ap
P.G. Burton and R.D. Brown, Approximute
proximate F matrix which has each element as close as possible to the corresponding exactly calculated F matrix element. The analysis suggested that new criteria were required for the selection of those integrals which made an essential contribution to the structure of the F matrix, that these could be found, and that possibilities existed for obtaining a reasonable approximation to the F matrix while still ignoring the larger proportion of the two electron multicentre integrals of the exact calculation. The purpose of this report is to describe in detail the general theory behind the development of new simplified LCAO SCF MO formalisms which are aimed at reproducing the essential structural elements (ESE) of the exact LCAO SCF MO F matrix. The ESE MO formalism described here is intended to form the basis for flexible, economical and above all reliable methods of electron structure calculation. We intend that these should feature completely theoretical parametrization. Our aim is to provide general methods suitable for use in invcstigations of electronic structure of large or low symmetry molecules or other atomic aggregates. Applicutions of the ESE MO formalism aimed initially at elucidating any limitations in the applicability of the formalism are the subject of a further report. Subseqtiently applications of the ESE MO formalism will be described which are aimed at developing a completely general prescription for theoretical parametrization of the formalism, with particular attention being paid to (i) basis set choice, (ii) methods of integral evaluation, and (iii) investigations of core/ valence separation approximations. Both open-shell (UHF) and closed shell variants of the ESE MO formalism are considered.
MO theory:
XF
=
UV
221
I
s
x;(I)fiWxJW.
S is the overlap matrix, s
flu
= j-x;WxJW
1
and t is the (diagonal [I 1) matrix of molecular orbital energies, Ei. Since xF depends on xC through the (closed shell) charge/bond-order matrix P, occ. hfos
C
‘A0 =2
‘ihCio
1
i
eq. (1) must be solved iterativelyt. Using ? for the kinetic energy operator,
and super-
scripts on the nuclear attraction operators, P, and subscripts on each orbital to denote the atomic centre io which the operator or orbital belongs, and with the two electron integral +1X0)=
s
~,(1?~(1)(1I’,~)x~(2)~6(3_)d~.
the explicit form for the one and two centrc matrix elements of xF are respectively XFlrAYA =
2. Theory
(c(,l++3A+gtiBl”*)
B+A
In the LCAO SCF method, a square matrix, XC, composed of columns of coefficients of each atomic orbital of the basis set, {x) = /.I, V, X, u . . . occurring in the expansion for each molecular orbital i, is obtained by solution of the matrix equation xFxC = SXCe. xF is the square Hartree-Fock over the chosen basis 1x1,
The ESE MO formalism.
+c A
+
BfA
C+A.B
2x
C(I
phBcrc [(~A~A(ABUC)-~~AOCI~~~~)I, (3
(I) hamiltonian
matrix t The generalization to open-shell systems using the UHF method of Popte snd Ncsbet [9] is trivial.
222
P.G. Burton and R.D. Brown, Approximated?0
theory: The ESE MO formalism. I
trality is a guide to the fact that interactions of an electron in the product distribution &) with the various nuclei of a molecule will be balanced piecewise to u kwge extent by the electrons associated with those nuclei. The analysis mentioned earlier [3] led to the recog nition that two electron integrals involving at most a single two centre (bicentric) orbital product distribution of the form (fiA uB) in integrals such as 01A pB Ih, or-), all C, were essentiul to any reasonable approximation to the overall F matrix, since these integrals occur in contributions to F of the form
c c
C CP&JA~BIXCUC) ’
C+A.B
C+A,B
F
c
C+A,B
DtA,B,C
-+ c h
c cl
a
~~oC[~AuBlhCUC)-IoAuCIAC~B)I
p+,D[bA”B~xCOD)
-1 b,
uDtxCvB)?. (3)
The element Fpy of the F matrix reflects the overall energy that a charge distribution arising from an electron in the distribution given by the product of the two orbitals x, and xy [denoted UP)] has as a .result of its kinetic energy, and the physical interactions of the electron with the nuclei and all the other electrons occupying various orbital produc! distributions @a). The effective population of each distribution (xu) is given by the appropriate bond order matrix element. The evaluation of the Fpy matrix elements includes allowance for the differing interaction of electrons of like, or of unlike spin, as prescribed in the HF theory. Using this interpretation of the form of the F matrix, the principle of electroneu-
u
Such contributions must be included since they represent the overall repulsive interaction between the ekCtrOIl charge distribution @A +$) and the (VdeI’ICe electrons associated with atom C. which by the or all) electroneutrality principle must be of comparable magnitude to tts attractive interaction of an electron in (&r$) with the central (con: or nuclear) positive charge of atom C. Integrals of this type are completely eliminated in all ND0 formalisms (including NDDO), since 011two electron integrals involving any bicentric orbital product distribution are eliminated. While the single bicentric distribution integrals are not required for a balanced estimation of one cenrre F matrix elements, they are essential for any balanced evaluation of the ho cenffe F elements by completely theoretical means [3]. The new approximate ESE MO formalism accordingly comprises full evaluation of all one electron integrals, inclusion of aU the NDDO level two electron integrals us well us the interactions of the form
that occur in the two centre F element over @A r$)The contributions to F included in the ESE formalism are those underlined in eqs. (2) and (3). AU the molecular integrals included in the ESE MO F matrix evaluation include at most a single bicenfric orbital product disnibutin (SBD). On the other hand, all the remaining omitted terms of the complete
P.G. Burton and RD. Brown. Approximate
Fmatrix either(i) involve the interaction of fwo bicentric orbital product distributions, or (ii) involve integrals with at least one bicentric distribution multiplied by a nvo cenrre, rather than a one centre, rather than a one centre, bond order matrix element (i.e., PhAq verSuS‘AA OA). If we associate a size reduction factor equivalent as an overlap element to unity (%A?. -1) with the size reduction in individual contributlons to the terms of F caused by the incidence of either a two centre bond order element (‘A/, U,$p$, 0 ), or a bicentric product distribution in an integral 4 fiAag):(XA aA)], then all the remaining terms in F omitted in the ESE formalism are of second or third order in overlap, while those included are of zero or first order in overlap. As the bond order elements and the multicentre two electron integrals occurring in the omitted terms of F would normally occur with either sign, it is possible that, in addition to being individually small, the overall effect of the neglected non-ESE terms of F would, through cancellation, be acceptably small even though a large number of integrals are involved. In marked contrast to the NDDO and less accurate NDO, and related, approximate MO formulations, the use of the ESE MO formulation as indicated in eq. (3) for evaluating two centre F matrix elements affords a consisrenr and an unbiased accounting of the detailed balance between stabilizing and destabilizing contributions to these elements. This allows the overall magnitude of the off diagonal F elements to be evaluated without making any assumptions about the nature of the bonding between the two atoms concerned in the two centre F matrix element, or the environment of those atoms. Since strict and consistent use of the ND0 integral approxirralions eliminates all but one minor two electron contribution to FPAQ elements, the value of these matrix elements must be parametrized in entirety in ND0 methods. Any parametrized two centre matrix elements can only be expected to be applicable when the AB fragment occurs in the molecule of interest in a similar environment and with similar bonding characteristics to the AB fragment in the system from which the parametrized FrAQ elements were obtained. The degree of mixing of orbitals centred on diffcrent atoms into molecular orbitals (i.e., covalency) depends on the magnitude of the connecting F,,A matrix elements. In view of this, in situations invo“B. vmg
MO theory: 77zeESE MO formlism
I
223
unusual bonding, or far investigation of molecules or series of molecules &owing a large variation in binding of distinct AB fnlgments, the completely flexible ESE MO formulation would appear to be more appropriate than any ND0 formulation (with its necessarily limited trunsfembifity of parametrization). A feature of the ESE MO formalism is that it is completely rotationally invariant to any local transformation of axes or of atomic centred basis functions, though all integml averaging is specifically avoided. This is essential if the very different characteristics of different orbitals of an atom are to be preserved and reflected in all stages of the wavefunction calculation. In calculations where higher valence orbitals (e.g., 3d orbitals for second row atoms) are included for example, which tend to be much more diffuse than the normal valence orbitals, it is important to take into account the differences in magnitudes of interactions in which these orbitals are iwolved relative to similar interactions involving normal valence orbit&, if any realistic estimate of the participation of higher valence orbitals in bonding is to be obtained. The arguments presented concerning the relative importance of contributions to the F matrix depend essentially on the magnitude of overlap integrals between basis orbit& centred on different nuclei of the molecule. it is important to note that there is no theorem or hard and fast rule for deciding when the ESE MO formalism can be reliably applied, and when it is not appropriate. It is of course essential to establish limits of applicability of the formalism, and such investigations will be the subject of subsequent reports. At this stage a number of observations can be made however. The limits of applicability of the ESE MO formalism will depend on the extent of oveclap of basis functions on different nuclei, so the nature of the basis set employed is a non-trivial consideration, particularly when diffuse functions are concerned. The ESE MO formalism can be expected to show lower performance as overlaps increase in short multiply bonded diatomic fragments of molecules and the extent of this must be established. Finally, a number of investigations (e.g., 14.51) have centred around the use of ND0 integral approximations applied in the symmetrically orthogonal basis of Liiwdin (61. The symmetric orthogonal&d basis is a multicentred basis set but each multicentre basis orbital generally has a predominant contribution from a
224
P.G. Burrorr and R.D. Brown, Apprvximare MO theory: The ESE
particular atomic orbital. In this sense therefore the neglect of integrals involving products of orbitats mainly associated with two different centres’has been considered, and the formulation of the ND0 methods in the L6wdin basis has the advantage that the magnitude of integrals containing differential overlaps (and therefore neglected in ND0 methods) tend to be smaller than corresponding atomic orbital basis integrals. Advantage could be taken of this characteristic of Ltiwdin basis integrals by fom-mlating the ESE MO formalism in the Lijwdin basis, so that all the neglected terms of the ESE MO formalism would be smaller, exceprfir the&r thul the integrals still required are integrals over multicentre orbitals, which would require a knowledge of (or approximation to) all the mdticentre integrals over atomic orbitals.
3. Numbers of integrals to be evaluated Relative to the ND0 methods, the ESE MO formalism obtains its generality and flexibility only by the retention of a small subset of additional integrals, but relative to ab initio calculations a very large number of integrals are still omitted. All four centre, most three centre and the two centre exchange type two electron integrals are all omitted by the ESE ht0 formalism. The reduction in the number of integrals which must be evahrated and stored is especially important in either very large molecules, or in molecules of low symmetry where very few integrals are equivalent or zero, and it is in these areas that the ESE MO formulation is anticipated to be most useful. In view of the desirability of reducing the integral evaluation and storage task, in MO calculations, it should be noted that within the ESE formalism so far discussed a further simplification is possible. The group of terms
occurring in the formula for the two centre F matrix elements (3) contains the only references to rhree cenfre integrals in the whole formalism. By reference to the very argurrents used previously for the i&~&n of the latter of these two terms, in the absence of
MO formalismI
local deviations from electroneutrality within the molecule. it could be inferred that this group of terms may have little overall effect on the v&e of the individual two centre F matrix elements. If this were the case, the group of terms (4) could be neglected entirely without detriment, thus saving the evaluation of all three centre integrals (both one- and two-electron) of the’ESE formalism. This latter scheme will be denoted single bicentric distributionfhuo centre only (SBD/X), while the complete scheme will be denoted single bicentric distribulion/three centre integrals included (SBD/3C). In table 1 we present a comparison of the numbers of integrals involved in (i) an ab initio calculation, (ii) an SBD/3C calculation (the full ESE MO formalism) and (iii) an SBD/X calculation. Table 1 shows clearly that both the SBD/3C and the SBD/X schemes represent a dramatic reduction in the integral evaluation and storage task of the ab initio calculation with the same basis, as the number of atoms (N), and the number of basis functions per atom (m), both increase.
4. Conclusion A large number of questions have been raised in the exposition of the ESE MO formalism embracing the SBD/3C and SBD/X! formulations. The enormous reduction in the effort required for integral evaluation storage and manipulation afforded by the SBD/3C and SBD/2C formulations particularly as the number of atoms and number of basis functions per atom increase provides the stimulus for investigation into the answer of those questions. One of us (PCB) has recently reviewed current approximate molecular orbital methods, particularly in relation
to inorganic
systems,
and that article records
much of the philosophy behind the present series of investigations into new simplified molecular orbital methods. An interesting development since that review was the PRDDO methods of Halgren and Lipscomb [8]. We note that those authors conclude that the @A ye IhA uA) integrals, which they neglect, are probably the key to successful parametrization of the PRDDO method. It was thought that no completely theoretical parametrization of their method, which lies intermediate in complexity between the ND0 and ab
P.G.
Bwfon
and
R.D.
Brown,
Approximate
MO
rheory:
nte
ESEMOformakn
r
225
Table 1 Numbers of distinct two-electron integrals required at various levelsof approximation in evaluation of the LCAO SCF F matrix. as a function of the numb& of atoms IV. each having m basis functions. The percentages of the cotal number of intcgraIs relative to the corresponding ab mitio formulation are recorded in brackets
iv
m 4
10
30
Ab initio
4 10 30
9.32 x 103(100) 3.37 x 10s (LOO) 2.64 X 10’ (100)
3.37 1.28 1.02
SBDl3C 1)
4 10 30
5.44 8.20 2.18
x 103 x IO4 x IO6
1.80 X IO5 (53.6) 2.78 X lo6 (21.8) 7.45 X IO’ (7.39)
1.35 x 10’ QI.2) 2.11 X 10’ (20.6) 6.28 X IO9 (6.16)
SE&l)
4 10 30
1.30 6.49 5.46
x 103 (14.0) x 103 (1.93) x lo4 (0.21)
4.08 X IO4 (12.1) 2.01 x 105 (1.57) 1.69 x lo6 (0.16)
2.99 x 106 (11.3) 1.46 X 10’ (1.43) 1.22 x 105 (0.12)
(59.5) (24.4) (8.26)
x 105 (100) X 10’ (100) x 109 (100)
2.64 x 10’ 1.02 x 109 1.02 x 10”
(100) (100) cKl0,
1) Seetext.
initio methods, was realizable without these integrals. This essentially bears out our conviction of the im-
portance of (abloa) type integrals, and helps justify their central role in the ESE MO formalism.
[3] R.D. Brown and P.C. Burton, Chem. Phys. Letters 20 (1973) 45. [4 1 R. McWeeny. Proc. Roy. Sot. A277 (1954) 288; D.B. Cook, P.C. Hallis and R. McWeeny. hfol. Phys. 13 (1967)553. (5 1 R.D. Brown. F.R. Burden and G.R. Williams, Theoret.
References
161 P.O. Lijwdin, J. Chem.
Chim. AcIa 18 (1970)
111 C.CJ. Roothaan, Rev. Mod. Phys. 23 (195 1) 69. 121 J.A. Pople, G.A. Segal and D.P. Sanlry. J. Chem. Phys. 43 (1965) S129; J.A. Pople and D.L. Beveridge, Approximale molecular orbital theory (hlffinw-Hill, New York, 1970).
98.
Phys.
18 (1950)
365.
171 P.G. Burton. Coord. Chem. Rev. 12 (1974) 37. [S] T.A. Halgren and W.N. Lipscomb. J. Chem. Phys. 58 (1973) 1569. [Y] J.A. Pople and R.K. Nest!&. J. Chcm. Phyr. 22 (1955)
571.
PHYSICS
4 (1974)
220-2X.0
NORTH-HOLLAND
PUBLISHING
COMPANY
APPROXIMATE MOLECULAR ORBITAL THEORY: THE ESE MO FORMALISM. I. GENERAL THEORY AND NOTATION P.G. BURTON Chemisrry
Department
t. Wollongong University
College, Wollongong, N.S. W. 2500, Australia
and R.D. BROWN Chemistry
Department.
hfonash University,
C&ytorr,
Victoria
3168.
Australta
Received 29 November 1973 Revised manuscript received 25 February 1974
A new formalism for simplified molecular orbital calculations is elucibted. ‘Ihe focus of the formalism is the of a good approximation to the LCAO SCF F matrix of Roothaan’s equations, and preoccupation with approximations to individual molecubr intepals is avoided. The geat majority of multicentre two-electron integrals of the exact formalism are found to be largely inconsequential to the attainment of a good approsimation to ihe F production
matrix.
I. Introduction Relatively littlc is known of the precise role of different classes of the multicentre integrals that occur in large numbers in the widely used LCAO SCF h10 appl&mation [I ] to the Hartree-Fock (HF) theory of molecular electronic structure calculation. The accuracy and reliability of any LCAO SCF MO calculation (relative to the “HF limit”) critically depends on the nature, extent and flexibility of the chosen basis set, in terms of which the molecular orbitals are expanded. However, as the size of the basis set is increased IO approach the HF limit, it is well known that the number of integrals over the expansion basis functions situated at different points in the molecule that must be evaluated, stored and manipulated increases dramatically. Many simplified LCAO SCF MO procedures have evolved which seek to reduce the complexity of an exact (ab initio) solution of the LCAO SCF MO method, by ignoring the contributions of some intet Permanent address to which all correspondence directed.
should be
grals that are either difficult to evaluate or that occur in inconveniently large numbers in the exact caiculation of the LCAO SCF wavefunction. The most widely used such methods have been based on the so-called CNDO and lND0 integral approximations, which together with the more complex NDDO approximation, constitute the general group of “neglect of differential overlap” (NDO) approximations [2]. Brown and Burton [3] have recently noted an analysis of the role that various classes of muiticentre two electron integrals play in the generation of the final SCF wavefunction. That analysis, based on simple physical and arithmetic arguments, casts strong doubt on the reliability and predictive capability of all approximate MO methods based on ND0 approximations. The basic premise of the analysis was that it is the shucfufe of the LCAO SCF F matrix (the size of the individual F matrix elements, their sign, and especially the rebfionship between different - particularly diagonal versus off-diagonal - F matrix elements) which determines the ultimate generation of covalency in the final wavefunction. Accordingly the aim of any simplified or approximate molecular orbital method must always be to produce at SCF convergence an ap
P.G. Burton and R.D. Brown, Approximute
proximate F matrix which has each element as close as possible to the corresponding exactly calculated F matrix element. The analysis suggested that new criteria were required for the selection of those integrals which made an essential contribution to the structure of the F matrix, that these could be found, and that possibilities existed for obtaining a reasonable approximation to the F matrix while still ignoring the larger proportion of the two electron multicentre integrals of the exact calculation. The purpose of this report is to describe in detail the general theory behind the development of new simplified LCAO SCF MO formalisms which are aimed at reproducing the essential structural elements (ESE) of the exact LCAO SCF MO F matrix. The ESE MO formalism described here is intended to form the basis for flexible, economical and above all reliable methods of electron structure calculation. We intend that these should feature completely theoretical parametrization. Our aim is to provide general methods suitable for use in invcstigations of electronic structure of large or low symmetry molecules or other atomic aggregates. Applicutions of the ESE MO formalism aimed initially at elucidating any limitations in the applicability of the formalism are the subject of a further report. Subseqtiently applications of the ESE MO formalism will be described which are aimed at developing a completely general prescription for theoretical parametrization of the formalism, with particular attention being paid to (i) basis set choice, (ii) methods of integral evaluation, and (iii) investigations of core/ valence separation approximations. Both open-shell (UHF) and closed shell variants of the ESE MO formalism are considered.
MO theory:
XF
=
UV
221
I
s
x;(I)fiWxJW.
S is the overlap matrix, s
flu
= j-x;WxJW
1
and t is the (diagonal [I 1) matrix of molecular orbital energies, Ei. Since xF depends on xC through the (closed shell) charge/bond-order matrix P, occ. hfos
C
‘A0 =2
‘ihCio
1
i
eq. (1) must be solved iterativelyt. Using ? for the kinetic energy operator,
and super-
scripts on the nuclear attraction operators, P, and subscripts on each orbital to denote the atomic centre io which the operator or orbital belongs, and with the two electron integral +1X0)=
s
~,(1?~(1)(1I’,~)x~(2)~6(3_)d~.
the explicit form for the one and two centrc matrix elements of xF are respectively XFlrAYA =
2. Theory
(c(,l++3A+gtiBl”*)
B+A
In the LCAO SCF method, a square matrix, XC, composed of columns of coefficients of each atomic orbital of the basis set, {x) = /.I, V, X, u . . . occurring in the expansion for each molecular orbital i, is obtained by solution of the matrix equation xFxC = SXCe. xF is the square Hartree-Fock over the chosen basis 1x1,
The ESE MO formalism.
+c A
+
BfA
C+A.B
2x
C(I
phBcrc [(~A~A(ABUC)-~~AOCI~~~~)I, (3
(I) hamiltonian
matrix t The generalization to open-shell systems using the UHF method of Popte snd Ncsbet [9] is trivial.
222
P.G. Burton and R.D. Brown, Approximated?0
theory: The ESE MO formalism. I
trality is a guide to the fact that interactions of an electron in the product distribution &) with the various nuclei of a molecule will be balanced piecewise to u kwge extent by the electrons associated with those nuclei. The analysis mentioned earlier [3] led to the recog nition that two electron integrals involving at most a single two centre (bicentric) orbital product distribution of the form (fiA uB) in integrals such as 01A pB Ih, or-), all C, were essentiul to any reasonable approximation to the overall F matrix, since these integrals occur in contributions to F of the form
c c
C CP&JA~BIXCUC) ’
C+A.B
C+A,B
F
c
C+A,B
DtA,B,C
-+ c h
c cl
a
~~oC[~AuBlhCUC)-IoAuCIAC~B)I
p+,D[bA”B~xCOD)
-1 b,
uDtxCvB)?. (3)
The element Fpy of the F matrix reflects the overall energy that a charge distribution arising from an electron in the distribution given by the product of the two orbitals x, and xy [denoted UP)] has as a .result of its kinetic energy, and the physical interactions of the electron with the nuclei and all the other electrons occupying various orbital produc! distributions @a). The effective population of each distribution (xu) is given by the appropriate bond order matrix element. The evaluation of the Fpy matrix elements includes allowance for the differing interaction of electrons of like, or of unlike spin, as prescribed in the HF theory. Using this interpretation of the form of the F matrix, the principle of electroneu-
u
Such contributions must be included since they represent the overall repulsive interaction between the ekCtrOIl charge distribution @A +$) and the (VdeI’ICe electrons associated with atom C. which by the or all) electroneutrality principle must be of comparable magnitude to tts attractive interaction of an electron in (&r$) with the central (con: or nuclear) positive charge of atom C. Integrals of this type are completely eliminated in all ND0 formalisms (including NDDO), since 011two electron integrals involving any bicentric orbital product distribution are eliminated. While the single bicentric distribution integrals are not required for a balanced estimation of one cenrre F matrix elements, they are essential for any balanced evaluation of the ho cenffe F elements by completely theoretical means [3]. The new approximate ESE MO formalism accordingly comprises full evaluation of all one electron integrals, inclusion of aU the NDDO level two electron integrals us well us the interactions of the form
that occur in the two centre F element over @A r$)The contributions to F included in the ESE formalism are those underlined in eqs. (2) and (3). AU the molecular integrals included in the ESE MO F matrix evaluation include at most a single bicenfric orbital product disnibutin (SBD). On the other hand, all the remaining omitted terms of the complete
P.G. Burton and RD. Brown. Approximate
Fmatrix either(i) involve the interaction of fwo bicentric orbital product distributions, or (ii) involve integrals with at least one bicentric distribution multiplied by a nvo cenrre, rather than a one centre, rather than a one centre, bond order matrix element (i.e., PhAq verSuS‘AA OA). If we associate a size reduction factor equivalent as an overlap element to unity (%A?. -1) with the size reduction in individual contributlons to the terms of F caused by the incidence of either a two centre bond order element (‘A/, U,$p$, 0 ), or a bicentric product distribution in an integral 4 fiAag):(XA aA)], then all the remaining terms in F omitted in the ESE formalism are of second or third order in overlap, while those included are of zero or first order in overlap. As the bond order elements and the multicentre two electron integrals occurring in the omitted terms of F would normally occur with either sign, it is possible that, in addition to being individually small, the overall effect of the neglected non-ESE terms of F would, through cancellation, be acceptably small even though a large number of integrals are involved. In marked contrast to the NDDO and less accurate NDO, and related, approximate MO formulations, the use of the ESE MO formulation as indicated in eq. (3) for evaluating two centre F matrix elements affords a consisrenr and an unbiased accounting of the detailed balance between stabilizing and destabilizing contributions to these elements. This allows the overall magnitude of the off diagonal F elements to be evaluated without making any assumptions about the nature of the bonding between the two atoms concerned in the two centre F matrix element, or the environment of those atoms. Since strict and consistent use of the ND0 integral approxirralions eliminates all but one minor two electron contribution to FPAQ elements, the value of these matrix elements must be parametrized in entirety in ND0 methods. Any parametrized two centre matrix elements can only be expected to be applicable when the AB fragment occurs in the molecule of interest in a similar environment and with similar bonding characteristics to the AB fragment in the system from which the parametrized FrAQ elements were obtained. The degree of mixing of orbitals centred on diffcrent atoms into molecular orbitals (i.e., covalency) depends on the magnitude of the connecting F,,A matrix elements. In view of this, in situations invo“B. vmg
MO theory: 77zeESE MO formlism
I
223
unusual bonding, or far investigation of molecules or series of molecules &owing a large variation in binding of distinct AB fnlgments, the completely flexible ESE MO formulation would appear to be more appropriate than any ND0 formulation (with its necessarily limited trunsfembifity of parametrization). A feature of the ESE MO formalism is that it is completely rotationally invariant to any local transformation of axes or of atomic centred basis functions, though all integml averaging is specifically avoided. This is essential if the very different characteristics of different orbitals of an atom are to be preserved and reflected in all stages of the wavefunction calculation. In calculations where higher valence orbitals (e.g., 3d orbitals for second row atoms) are included for example, which tend to be much more diffuse than the normal valence orbitals, it is important to take into account the differences in magnitudes of interactions in which these orbitals are iwolved relative to similar interactions involving normal valence orbit&, if any realistic estimate of the participation of higher valence orbitals in bonding is to be obtained. The arguments presented concerning the relative importance of contributions to the F matrix depend essentially on the magnitude of overlap integrals between basis orbit& centred on different nuclei of the molecule. it is important to note that there is no theorem or hard and fast rule for deciding when the ESE MO formalism can be reliably applied, and when it is not appropriate. It is of course essential to establish limits of applicability of the formalism, and such investigations will be the subject of subsequent reports. At this stage a number of observations can be made however. The limits of applicability of the ESE MO formalism will depend on the extent of oveclap of basis functions on different nuclei, so the nature of the basis set employed is a non-trivial consideration, particularly when diffuse functions are concerned. The ESE MO formalism can be expected to show lower performance as overlaps increase in short multiply bonded diatomic fragments of molecules and the extent of this must be established. Finally, a number of investigations (e.g., 14.51) have centred around the use of ND0 integral approximations applied in the symmetrically orthogonal basis of Liiwdin (61. The symmetric orthogonal&d basis is a multicentred basis set but each multicentre basis orbital generally has a predominant contribution from a
224
P.G. Burrorr and R.D. Brown, Apprvximare MO theory: The ESE
particular atomic orbital. In this sense therefore the neglect of integrals involving products of orbitats mainly associated with two different centres’has been considered, and the formulation of the ND0 methods in the L6wdin basis has the advantage that the magnitude of integrals containing differential overlaps (and therefore neglected in ND0 methods) tend to be smaller than corresponding atomic orbital basis integrals. Advantage could be taken of this characteristic of Ltiwdin basis integrals by fom-mlating the ESE MO formalism in the Lijwdin basis, so that all the neglected terms of the ESE MO formalism would be smaller, exceprfir the&r thul the integrals still required are integrals over multicentre orbitals, which would require a knowledge of (or approximation to) all the mdticentre integrals over atomic orbitals.
3. Numbers of integrals to be evaluated Relative to the ND0 methods, the ESE MO formalism obtains its generality and flexibility only by the retention of a small subset of additional integrals, but relative to ab initio calculations a very large number of integrals are still omitted. All four centre, most three centre and the two centre exchange type two electron integrals are all omitted by the ESE ht0 formalism. The reduction in the number of integrals which must be evahrated and stored is especially important in either very large molecules, or in molecules of low symmetry where very few integrals are equivalent or zero, and it is in these areas that the ESE MO formulation is anticipated to be most useful. In view of the desirability of reducing the integral evaluation and storage task, in MO calculations, it should be noted that within the ESE formalism so far discussed a further simplification is possible. The group of terms
occurring in the formula for the two centre F matrix elements (3) contains the only references to rhree cenfre integrals in the whole formalism. By reference to the very argurrents used previously for the i&~&n of the latter of these two terms, in the absence of
MO formalismI
local deviations from electroneutrality within the molecule. it could be inferred that this group of terms may have little overall effect on the v&e of the individual two centre F matrix elements. If this were the case, the group of terms (4) could be neglected entirely without detriment, thus saving the evaluation of all three centre integrals (both one- and two-electron) of the’ESE formalism. This latter scheme will be denoted single bicentric distributionfhuo centre only (SBD/X), while the complete scheme will be denoted single bicentric distribulion/three centre integrals included (SBD/3C). In table 1 we present a comparison of the numbers of integrals involved in (i) an ab initio calculation, (ii) an SBD/3C calculation (the full ESE MO formalism) and (iii) an SBD/X calculation. Table 1 shows clearly that both the SBD/3C and the SBD/X schemes represent a dramatic reduction in the integral evaluation and storage task of the ab initio calculation with the same basis, as the number of atoms (N), and the number of basis functions per atom (m), both increase.
4. Conclusion A large number of questions have been raised in the exposition of the ESE MO formalism embracing the SBD/3C and SBD/X! formulations. The enormous reduction in the effort required for integral evaluation storage and manipulation afforded by the SBD/3C and SBD/2C formulations particularly as the number of atoms and number of basis functions per atom increase provides the stimulus for investigation into the answer of those questions. One of us (PCB) has recently reviewed current approximate molecular orbital methods, particularly in relation
to inorganic
systems,
and that article records
much of the philosophy behind the present series of investigations into new simplified molecular orbital methods. An interesting development since that review was the PRDDO methods of Halgren and Lipscomb [8]. We note that those authors conclude that the @A ye IhA uA) integrals, which they neglect, are probably the key to successful parametrization of the PRDDO method. It was thought that no completely theoretical parametrization of their method, which lies intermediate in complexity between the ND0 and ab
P.G.
Bwfon
and
R.D.
Brown,
Approximate
MO
rheory:
nte
ESEMOformakn
r
225
Table 1 Numbers of distinct two-electron integrals required at various levelsof approximation in evaluation of the LCAO SCF F matrix. as a function of the numb& of atoms IV. each having m basis functions. The percentages of the cotal number of intcgraIs relative to the corresponding ab mitio formulation are recorded in brackets
iv
m 4
10
30
Ab initio
4 10 30
9.32 x 103(100) 3.37 x 10s (LOO) 2.64 X 10’ (100)
3.37 1.28 1.02
SBDl3C 1)
4 10 30
5.44 8.20 2.18
x 103 x IO4 x IO6
1.80 X IO5 (53.6) 2.78 X lo6 (21.8) 7.45 X IO’ (7.39)
1.35 x 10’ QI.2) 2.11 X 10’ (20.6) 6.28 X IO9 (6.16)
SE&l)
4 10 30
1.30 6.49 5.46
x 103 (14.0) x 103 (1.93) x lo4 (0.21)
4.08 X IO4 (12.1) 2.01 x 105 (1.57) 1.69 x lo6 (0.16)
2.99 x 106 (11.3) 1.46 X 10’ (1.43) 1.22 x 105 (0.12)
(59.5) (24.4) (8.26)
x 105 (100) X 10’ (100) x 109 (100)
2.64 x 10’ 1.02 x 109 1.02 x 10”
(100) (100) cKl0,
1) Seetext.
initio methods, was realizable without these integrals. This essentially bears out our conviction of the im-
portance of (abloa) type integrals, and helps justify their central role in the ESE MO formalism.
[3] R.D. Brown and P.C. Burton, Chem. Phys. Letters 20 (1973) 45. [4 1 R. McWeeny. Proc. Roy. Sot. A277 (1954) 288; D.B. Cook, P.C. Hallis and R. McWeeny. hfol. Phys. 13 (1967)553. (5 1 R.D. Brown. F.R. Burden and G.R. Williams, Theoret.
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