Atomic Orbitals from Molecular Wave Functions: The Effective

Standard methods give for the coefficients the Hermitian n by n matrix eigenvalue equation As the eigenvalues of an Hermitian matrix form a complete ...
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J. Phys. Chem. 1996, 100, 6249-6257

6249

Atomic Orbitals from Molecular Wave Functions: The Effective Minimal Basis I. Mayer Central Research Institute for Chemistry of the Hungarian Academy of Sciences, H-1525 Budapest, P.O. Box 17, Hungary ReceiVed: September 20, 1995; In Final Form: February 7, 1996X

The recent idea of extracting effective atomic orbitals from molecular wave functions by performing independent localization transformations for each atom separately is generalized to the case of an arbitrary Hermitian bilinear localization functional. The general equations are derived and the orthogonality relationships pertinent to the localized molecular orbitals are proved. The “intraatomic components” of these localized orbitals form an effective atomic basis, which is also automatically orthogonal if some conditions are fulfilled. Several different localization functionals are considered and it is shown that for the simplest one the orbitals obtained are natural hybrids in McWeeny’s sense and are conceptually close to (but not identical with) Weinhold’s “natural hybrid orbitals”. In this case one obtains for each atom of a “usual” molecule as many effective AOs of appreciable importance as the number of orbitals contained in the classical “minimal basis” of that atom, forming therefore asdistorted but still orthogonalseffective minimal basis of the atom within the molecule. The similarities and differences with Weinhold’s “atomic natural orbitals” are also discussed. It is pointed out that by selecting a proper localization functional, the present approach can also be used to define the effective atomic orbitals in a basis-free manner, i.e. even if no atom-centered basis was used in calculating the wave function. A possibility of generalization for correlated wave functions is also suggested.

1. Introduction Our qualitative understanding of molecular structure strongly relies on the notion that the atoms enter the molecules with their 1s, 2s, 2p, etc. orbitals (or their hybrids). It is not, however, a trivial task to find proper connections between this picture and the results obtained in the large scale ab initio calculations using extended basis sets. The “natural hybrid orbitals”, “natural bond orbitals” (NBOs) and “natural atomic orbitals” (NAOs)1-3 of Weinhold’s group represent significant achievements in this respect. The basic idea behind them is to obtain effective atomic or bond orbitals by diagonalizing the respective atomic or diatomic blocks of the first-order density matrix. This goes back to the classical work of McWeeny.4 Recently5,6 we have started to approach this problem from a quite different, specific point of view, which can be formulated as follows. Let us consider a single determinant wave function andsinstead of the canonic onessuse some localized molecular orbitals (LMOs), each of which is chosen to be maximally localized on one of the atoms. An LMO with a considerable weight on the atom in question can then be divided into an intraatomic part and an external one; the intraatomic part can then be considered as an effectiVe atomic orbital (AO) within the molecule. Obviously, these AOs contain much of information about the actual state of the atom in the given chemical environment. As the conservation of the orthogonality of the molecular orbitals (MOs) used to build up the determinant wave function is only a matter of convenience but not that of essence,7 we do not require its conservation, and allow the orbitals to be nonorthogonal, provided that this permits to increase the degree of their localization. However, postulating the nonorthogonality of LMOs is only important to justify that the orbitals localized on different centers are determined independently; it does not explicitly enter the formalism. In the previous papers we have considered two specific cases, differing by the localization criterion used to determine the X

Abstract published in AdVance ACS Abstracts, March 15, 1996.

0022-3654/96/20100-6249$12.00/0

LMOs: in ref 5 we used the Magnasco-Perico criterion8 derived from Mulliken’s population analysis in which one maximizes Mulliken’s net atomic population produced by the LMO on the selected atom, while in ref 6 the relative weight of the orbital in the given atomic domain corresponding to Bader’s topological theory9 of atoms was maximized. In both cases it was found that the localized orbitals corresponding to the same atom are automatically orthogonal and have orthogonal “atomic truncations”. The numerical calculations5 indicated that for the “ordinary” compounds there are as many molecular orbitals appreciably localizable on the given atom as the number of orbitals in the classical “minimal basis” for that atom. In this manner one can extract from the molecular wave function thesdistorted but still orthogonalseffectiVe minimal basis by which the given atom participates in forming the molecule. The conceptual importance of such a conclusion is that it represents a strong corroboration of the classical LCAO picture and indicates the octet rule being inherent in the results of the largescale ab initio calculations.5 It will be proved in the present paper that the AO-s obtained in this manner (i.e by using the Mulliken’s criterion) are natural hybrids in McWeeny’s sense4 and are close to (but not identical with) the “natural hybrid orbitals” of Weinhold and coworkers1-3 for the core and lone pair orbitals.10 For other electrons these authors use hybrid AOs defined as intraatomic parts of the NBOs derived, in turn, from the analysis of diatomic blocks of the density matrix. (An orthogonalization step is also included to obtain the effective minimal basis.) It is our opinion, however, that one can avoid arbitrariness only if all the orbitals of a given atom are obtained “at once”, by solving the same, common equation and that all the atomic orbitals with significant occupation numbers have an independent interest and their analysis can promote our understanding of the electronic structure of different molecules. A project probably more closely related to ours was performed by Weinhold’s NAO analysis.2,3 However, Weinhold’s NAOs are all orthogonal, while in our approach the interatomic overlap is conserved. We © 1996 American Chemical Society

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Mayer

consider overlap between the orbitals of different atoms extremely important from a genuine chemical point of view; it is significant for understanding both the chemical bonding and the repulsion of nonbonded atoms. (The relationships of our orbitals and Weinhold’s NAOs and “pre-NAO”s will also be discussed below in some detail.) It has been found in ref 6 that there were many similarities in the deviation with those in ref 5, although the localization criteria used were conceptually different (one working in the Hilbert space, another in the three-dimensional “physical” one). Accordingly, in this paper we present a quite general, abstract formulation, which can easily be adapted to any localization criterion which can be formulated through a Hermitian bilinear functional and introduce a few new functionals, too. As already noted, we shall show that the effective AOs obtained by our procedure represent natural hybrids and this is the point where our scheme touches that of Weinhold’s group. The connection with natural hybrids is very important also because it opens the way of generalizing the present concepts to correlated wave functions. When the “topological criterion” is used,6 then the effective AOs can be obtained without any explicit reference to the basis orbitals used in the calculations; this means that the concept of effective AOs can be, at least in principle, generalized to the Hartree-Fock limit or even to the exact wave function.11 We shall also discuss briefly another class of functionals making such a generalization possible without introducing discontinuities in the resulting orbitals. Some numerical examples will also be shown for the case of three different functionals.

where

Qjki ) Fi(φjc;φck)

(7)

are the elements of the Hermitian n by n matrix Qi and ai is the vector formed of the coefficients aij. Standard methods give for the coefficients aij the Hermitian n by n matrix eigenvalue equation n

Qkjiaji ) Miaik ∑ j)1

(8)

As the eigenvalues of an Hermitian matrix form a complete orthonormal basis, this result means that the condition eq 4 defines a unitary transformation of the occupied orbitals for the case of each Fi (i.e., for each atom). Accordingly, the parameters Mi are necessarily real. 2.2. Alternative Equations. In practice one uses finite basis expansions: m

m

c χµ, φjc ) ∑ cj,µ

ψi ) ∑ ei,µχµ

µ)1

(9)

µ)1

and has to determine the coefficients ei,µ of the orbital ψi. This can be done through the quantities aij above: n

c ei,µ ) ∑ajicj,µ

(10)

j)1

2. General Formalism 2.1. Basic Equations. Let Fi(φ;ψ) be a Hermitian bilinear functional12 of the one-electron functions (orbitals) φ,ψ, which means that

Fi(φ1;φ2) ) [Fi(φ2;φ1)]*

(1)

and

Fi(Rφ1 + βφ2;ψ) ) R*Fi(φ1;ψ) + β*Fi(φ2;ψ) Fi(φ;Rψ1 + βψ2) ) RFi(φ;ψ1) + βFi(φ;ψ2)

δei ) ηPSd (2)

and let us consider the Rayleigh-type ratio (expectation value) for the case of φ ) ψ ) ψi:

Mi )

Fi(ψi;ψi)

(3)

〈ψi|ψi〉

Now, we shall search for a maximal (or, at least, stationary) value of Mi:

δMi ) δ

[

]

Fi(ψi;ψi) 〈ψi|ψi〉

)0

Alternatively, one can search for the solutions of the stationary problem directly in terms of the coefficients ei,µ: When considering the variation in eq 4, the coefficients ei,µ may not be varied completely freely, because the orbital ψi must remain in the subspace of the occupied SCF orbitals even after the variation is performed. The most general permitted variation of the m-dimensional vector ei formed of the coefficients ei,µ is, therefore, the projection

(4)

restricting ψi to lie entirely in the occupied one-electron subspace (ψi ∈ {occ}). This means that ψi is a linear combination of the n occupied canonic orbitals φcj :

(11)

where d is a quite arbitrary m-dimensional vector, η is a (generally complex) variation parameter tending to zero and n

P ) ∑cjccjc†

(12)

j)1

is the usual LCAO “density matrix”. Performing the variation of the fraction in eq 4, we get the stationarity condition in the form

Fi(δψi;ψi) - Mi 〈δψi|ψi〉 + c.c. ) 0

(13)

where c.c. denotes the complex conjugate. Substituting eq 11 into eq 13 and taking into account that η contains an arbitrary phase factor, due to which the expression written down explicitly in the left-hand side of eq 13 and its complex conjugate should vanish separately, we get

n

ψi ) ∑ajiφjc

(5)

d†SPFiei ) Mid†SPSei

(14)

j)1

where Fi is the m by m matrix with the elements

Then the stationary condition can be written as

[ ]

ai†Qiai )0 δMi ) δ ai†ai

i ) Fi(χµ;χν) Fµν

(15)

(6) As d is an arbitrary vector and we may assume that the overlap

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J. Phys. Chem., Vol. 100, No. 15, 1996 6251 the orbital ψi. The localization functional is defined through its matrix:

matrix S is not singular, eq 14 reduces to

PFiei ) MiPSei

(16)

Because ψi ∈ {occ}, PSei ) ei. Using this, we get for all solutions with Mi * 0:

PFiei ) Miei

Fi ) LiSLi

is the “cut-off” matrix, the block of which correwhere sponding to the basis functions of the given atom A is a unit matrix and all other elements are zero:

{

(17) i ) Lµν

or

(22)

Li

if µ, ν ∈ A otherwise

δµν 0

(23)

This means that the elements of the matrix Fi are defined as

SPFiPSei ) MiSei

(18)

Equations 17 and 18 represent an eigenvalue equation of a nonHermitian matrix and a generalized eigenvalue equation of a Hermitian matrix, respectively. It can be shown, that in all cases with Mi * 0 eq 17 and 18 are equivalent with eq 16 and with eq 8 and give solutions which automatically lie in the occupied subspace. 2.2. Orthogonality Properties. It follows from the Hermiticity of the general equations eq 8 that the solutions with (k) different Mi values M(j) i and Mi are automatically orthogonal

(Mi(j) - Mi(k))ei(j)†Sei(k) ) 0

(19)

(In the degenerate case they may be chosen as such.) This is equivalent with the statement above related to the unitary character of the transformation diagonalizing the Hermitian matrix Qi. Another interesting property can be obtained from eq 17. We write down this equation for the solution e(k) i and the adjoint utilizing the Hermiticity of equation for the solution e(j) i matrices Fi and P:

i ) Fµν

{

(20)

(As noted above, the eigenvalues Mi are always real.) Now, i we multiply the first of these equations by e(j)† i F from the left i (k) and the second by F ei from the right, subtract the first equation from the second one and get

(Mi(j) - Mi(k))ei(j)†Fiei(k) ) 0

(21)

If matrix Fi is a positive definite one, then it can be considered to define a metrics, and eq 21 gives a new type of orthogonality relationships.13 As will be seen, in most cases interesting for us the matrix Fi can be assigned the meaning of the intraatomic oVerlap matrix.

3. Definition of the Effective Atomic Orbitals 3.1. Actual Forms of the Functional Fi. As noted in the introduction, we have previously considered two types of functional Fi (cases A and B below); a few others, also of population type, are added here. The localization functionals are distinguished by their “target functions”. A. Mulliken’s Net Population. In this case5 one maximizes Mulliken’s net population on the given atom, corresponding to

(24)

According to eqs 22 and 24 the ratio Mi is simply

∑

Mi )

e* i,µSµνei,ν

µ,ν∈A

e* ∑ i,µSµνei,ν µ,ν

(25)

B. Bader’s Population. In this case6 the relative weight of the orbital ψi within the atomic domain defined in the sense of Bader’s topological theory of atoms9 is maximized. The respective localization functional contains integration over the atomic domain ΩA:

r b) r dV Fi(φ;ψ) ) ∫Ω φ*(b)ψ(

(26)

i A ) Sµν ) ∫Ω χ*µ(b)χ r ν(b) r dV Fµν

(27)

A

leading to A

and

PFiei(k) ) Mi(k)ei(k) ei(j)†FiP ) Mi(j)ei(j)†

if µ, ν ∈ A otherwise

Sµν 0

Mi )

A e* ∑ i,µSµνei,ν µ,ν

e* ∑ i,µSµνei,ν µ,ν

(28)

C. The Projection on the Atomic Subspace. One searches for the MOs ψi for which the projection on the subspace spanned by the basis orbitals of the selected atom has a maximal, or at least stationary, norm (Roby’s population14). In this case the localization functional represents a matrix element (integral) of the operator of projection Pˆ A on the atomic subspace:

r ˆ Aψ(b) r dV Fi(φ;ψ) ) ∫φ*(b)P

(29)

Operator Pˆ A can be most conveniently defined in Dirac’s “Bra” and “Ket” notations:

Pˆ A )

∑ |χF〉 (SA-1)Fτ 〈χτ|

(30)

F,τ∈A

SA-1 being the inverse of the intraatomic block of the overlap matrix. When writing eq 29, it was explicitly utilized that operator Pˆ A is idempotent:

(Pˆ A)2 ) Pˆ A

(31)

i is In this case the matrix element Fµν

i ) Fµν

SµF(SA-1)FτSτν ∑ F,τ∈A

(32)

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Mayer

and the ratio Mi is obtained as -1 e* ∑ ∑ i,µSµF(SA )FτSτνei,ν µ,ν F,τ∈A

Mi )

(33)

∑e*i,µSµνei,ν µ,ν

D. Mulliken’s Gross Populations. One may also require Mulliken’s gross atomic population on the selected atom, originating from the orbital ψi to be stationary; this requirement slightly resembles the Pipek-Mezey localization criterion.15 Mulliken’s gross atomic population qA on atom A corresponding to a normalized orbital φ of either spin with the expansion coefficients forming vector c can be written as16

qA )

(cc†S)µµ ) ∑ ∑cµc* ∑ νSνµ µ∈A µ∈A ν

(34)

However, in order to obtain a Hermitian functional, we have used Pipek’s17 symmetrized form valid also in the complex case. In our notations it can be written as

qA )

1

[(cc†S)µµ + (Scc†)µµ] ) ∑ 2µ∈A 1

∑ ∑ν (cµc*νSνµ + Sµνcνc*µ)

2µ∈A

(35)

Accordingly, if the expansion coefficients of orbitals φ and ψ form vectors c and d, respectively, the Hermitian localization functional Fi(φ;ψ) can be defined as

1

Fi(φ;ψ) )

∑ [(cd S)µµ + (Scd )µµ] †

†

2µ∈A

amplitudes in the vicinity of several atoms, so they are lacking any well-defined atomic character. Of course, the functional using Bader’s population represents the ultimate remedy in these cases. Bader’s analysis is, however, a relatively complicated and costly numerical procedure and has also the disadvantage that it leads to effective AOs with discontinuities: each AO is defined only within the respective atomic domain so ends abruptly at its bordering surface. This also means that there is no overlap population in this case, and the chemically useful concept of bonds formed by strongly overlapping orbitals is also lost. A way out in this situation may be the introduction of a functional containing a “switch-off” weight function wi(r b b- B RA), depending on the distance (and, possibly, -B RA) ) f2i (r relative orientation) with respect to the nucleus of atom A. (The second form with the square has the advantage that it stresses the nonnegative character of the weight function and it will also be convenient in the latter derivations.) We shall not specify b- B RA); it is expected here the detailed form of the function fi(r to equal (or practially equal) unity in the internal atomic region and to fall gradually to zero at the distances larger than, say, the covalent radius of the atom. Obviously, a special case of such a switch-off function is used above where the target b- B RA) ) 1 function is Bader’s population - in that case fi(r within the atomic domain and zero outside. In this case one can write

(36)

Fi(φ;ψ) ) ∫φ*(b)f r i2(b r -B RA)ψ(b) r dV

(40)

i A ) Sµν ) ∫ χ*µ(b)f r i2(b r -B RA)χν(b) r dV Fµν

(41)

and

Mi )

In scalar form eq 36 can be written as

Fi(φ;ψ) )

1

∑ ∑(cµd*νSνµ + Sµνcνd*µ) 2µ∈A ν

(37)

(38)

and the contribution of the orbital ψi to Mulliken’s gross atomic population on atom A is

1 Mi )

∑ ∑ν (ei,µe*i,νSνµ + Sµνei,νe*i,µ)

2µ∈A

e*i,µSµνei,ν ∑ µ,ν

∑e*i,µSµνei,ν

(42)

µ,ν

Matrix Fi is then, by using matrix Li already defined in eq 23,

Fi ) 1/2(LiS + SLi)

A e* ∑ i,µSµνei,ν µ,ν

(39)

E. Use of a “Switch-Off” Function. The above functionals (except that using Bader’s population) are based on the assumption that the wave functions were obtained by using an atom-centered basis set. This assumption may be restrictivese.g., it excludes the cases when off-centered basis orbitals (bond functions) are used, and can become problematic conceptually in the limit of Very large basis sets: all the above schemes (except B) must fail when the basis of one atom becomes sufficiently complete to give an (even poor) approximation to the orbitals of the neighboring one. Numerical problems may be anticipated even for relative poor basis sets if they contain diffuse functions; the latter decay slowly and have comparable

The last equation looks formally the same as in the Bader case. 3.2. The Effective Atomic Orbitals. After performing the localization procedure for atom A, there is a meaning to separate out the intraatomic part ψAi of the localized orbital ψi obtained. The proper definition of this intraatomic component should be, of course, in accord with the localization functional Fi applied. In cases A and D above (i.e when Mulliken’s net and gross populations are used as target functions), the intraatomic part of the LMO ψi can be defined as its “intraatomic truncation”, i.e., that part of the LMO, which is obtained by conserving only the atomic basis functions in the expansion of ψi:

ψiA )

∑ ei,µχµ

(43)

µ∈A

while in the Bader case (case B above) the truncation should be performed in the physical space, i.e by conserving that part of ψi which is within the atomic domain ΩA:

ψiA(b) r )

{

ψi(b) r 0

if b r ∈ ΩA otherwise

(44)

In the case of the projective localization (case C above) the situation is slightly more complicated. It is most natural to define the intraatomic part of ψi as its projection

ψiA ) Pˆ Aψi

(45)

which is, in accord which eq 30, a linear combination of the basis orbitals centered on atom A, but does not coincide with

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J. Phys. Chem., Vol. 100, No. 15, 1996 6253

the “intraatomic truncation” of ψi which one could obtain in the spirit of eq 43. The latter may be of some interest, too, in analyzing the structure of the MOs. If the “switch-off” function is applied in the localization (case E above) then the appropriate definition of the “intraatomic part” of the orbital ψi is

ψiA(b) r ) fi(b r -B RA)ψi(b) r

(46)

This is the reason why we have introduced the weight function b- B RA). as a square of fi(r According to the definitions given in eqs 24, 27, 32, and 41, in all cases, except D, matrix Fi can be considered as the intraatomic overlap matrix,18 so it follows from eq 21 that the intraatomic components are orthogonal to each other (or can be selected so in the degenerate case) and the eigenvalues Mi give the square of the norm of the intraatomic part ψAi (provided that ψi is normalized to unity):

Mi )

〈ψiA|ψiA〉 〈ψi|ψi〉

(47)

The above results indicate that the requirement δMi ) 0 not only generates its “own” unitary transformation of the occupied orbitals for each atom (and each localization criterion), but one can classify the orbitals obtained according to the eigenvalues Mi: if Mi is large, then the orbital ψi is localized on the atom to a significant degree, while a zero (or very small) Mi value indicates that the given MO is located completely (mainly) “outside” of the atom considered. It seems natural to consider the intraatomic parts ψAi of the orbitals ψi having nonzero (nonnegligible) Mi values as the effectiVe atomic orbitals (hybrids) within the molecule. They can be renormalized to unity as

χiA )

1

xMi

ψiA

(48)

to get an orthonormalized set. According to eq 48, the coefficient of the normalized atomic orbital (hybrid) χAi in the MO ψi is xMi, so the eigenvalue Mi gives a measure of the extent to which orbital ψi is localized on the atom considered in this sense, too. (The orbitals with Mi ) 0 do not have any intraatomic components, at all.) These considerations do not apply in the case D above (localization using the Mulliken’s gross population criterion) because matrix Fi in eq 38 is not, in general, positive (semi)definite. However, the effective AOs can be obtained by renormalizing the “atomic truncations” in this case, toosbut they will not, in general, form an orthogonal set. (In this case the eigenvalues Mi are not related to any norm but give the contribution of the orbital ψi to Mulliken’s gross atomic population on atom A; this contribution can also be negative. If the projection criterion is used (case C), the effective atomic orbitals can be shown to have some interesting “pairing” properties: the orbitals ψi and χAi form the sets of the “corresponding orbitals”19-21 for the occupied MOs from one side and the atomic basis of atom A from another. In cases A, C, and D the orbitals χAi represent linear combinations of the original AO basis orbitals. If the Bader’s criterion or the “switch-off function” are used (cases B and E, respectively), the above formalism permits us to define the effective AOs in the molecule in a basis-independent fashion; in principle, one could extract the effective AOs even from the exact Hartree-Fock wave functions (obtained, e.g., by solving

the HF equations numerically). This extends the applicability of the LCAO concept. 3.3. The Number of Effective AOs and the Effective Minimal Basis. There are two considerations giving limits to the possible number of orbitals with Mi * 0, i.e. of the effective AOs located on the given atom. First, in cases A and C discussed above, their number cannot exceed the number mA of basis orbitals on the given atom A. This follows from the fact that the orbitals χAi are linear combinations of the original basis orbitals χµ; µ ∈ A and one cannot form more than mA orthogonal (or, more generally, linearly independent) combinations of them.22 Another, more important, observation is that the number of effective AOs participating in forming the molecular wave function cannot exceed the number of occupied MOs; this simply follows from the procedure by which one determines them by performing a unitary transformation of the occupied MOs and taking the intraatomic components of the n new orbitals obtained.23 This conclusion does not depend on the localization functional Fi applied. Now, let us consider the case of first row hydrides CH4, NH3, H2O, and HF. In each of these cases there are five doubly occupied orbitals, so one gets on the central atom five effective AOs, exactly as many as is the number of the classical minimal basis orbitals on these atoms. (Also, there is only one effective AO on each atom in the H2 molecule, and up to two in LiH, etc.) This conclusion is independent of how large the basis used is, and remains valid even in the case if no basis expansion technique is applied, provided that the functional B or Esor some analogous onesis used. Quite similar considerations hold for the second row hydrides (SiH4, PH3, H2S, HCl), only in their case the number of effective atomic orbitals is nine and not five. (And so on, for the higher main-group elements.) This means that the electronic structure of these molecules can exactly be treated in the closed-shell SCF theory (up to the HartreeFock limit) by postulating asdistorted but still orthogonals effectiVe minimal basis by which these atoms participate in forming the molecular wave function. (Of course, this effective minimal basis can be determined only a posteriori, by solving the equations discussed above. It may happen, in principle, that not all of these orbitals have a physical significance. Note that for electron-deficient molecules like BH a part of the “classical” minimal basis orbitals is empty and is not recovered, therefore, by the procedure of eq 8 based on transforming the canonic orbitals.) No similar considerations apply for more complicated molecules, nor even for the hydrogen atoms in the simple hydrides discussed above.24 However, complicated molecules often may be at least formally, “derived” by step-by-step substitution from the simple hydrides, and one can expect, that in many cases the electronic structure of the atom does not change so drastically from molecule to molecule, as to lead to qualitatiVe deviations from the above behavior. In other words, one may hope to find a proper localization functional which for mostsif not alls“usual” molecules (e.g. molecules formed from the main group elements only, etc.) will lead to as many effective AOs of appreciable importance (i.e., having a not too small Mi value) on each atom, as many orbitals are contained in the classical “minimal basis” for that atom. The set of orbitals obtained in this manner can again be called the effectiVe minimal basis of the atom in the molecule. We shall show in section 5 some examples indicating that functionals A and D really give such results. No such behavior can be expected in the case of the projective functional C, because even the bonding orbitals of vicinal bonds may have a quite significant overlap. (Functionals B and E have not yet been tested numerically.)

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Mayer

4. The Effective AOs as Natural Hybrids 4.1. Relations to McWeeny’s Natural Hybrids (Case A). Natural orbitals were defined25 as orbitals diagonalizing the oneb;r b′). (We consider the spinless case electron density matrix F1(r only.) By analogy, McWeeny defined4 “natural hybrids” as a transformed set of atomic basis orbitals diagonalizing the intraatomic portion of F1 within the molecule. Doing such a diagonalization in practical calculations, one has to keep in mind that the finite basis representation of the first order density matrix F1 is quite straightforward in orthonormal basis (McWeeny considered such a case4) but has some complications if the basis is oVerlapping. It has been shown26 that in the overlapping case the correct LCAO representation of the first order density matrix (density operator) is not matrix P, but the (nonHermitian) matrix PSsthe latter has the correct trace, projection, and idempotency properties, and the natural orbitals can also be calculated as its eigenvectors.27 Accordingly, one expects that the proper natural hybrids can be obtained by diagonalizing the matrix PASA, where PA and SA are the intraatomic portions of matrices P and S, respectively.28 By introducing an auxiliary orthonormalized basis on the atom in question, one can easily see that the orbitals obtained in this manner coincide with McWeeny’s natural hybrids,4 and no actual diagonalization of any non-Hermitian matrix should be accomplished. The immediate connections between our formalism and the natural hybrids can be seen by considering eq 17 and observing that matrix Fi behaves as an intraatomic overlap matrix for most of the functionals we have discussed above. In particular, in the simplest case A based on Mulliken’s net atomic population, the single nonzero block of matrix Fi coincides with the intraatomic block SAscf. eq 24. This means that in case A above the effective atomic orbitals χAi coincide with the McWeeny’s “natural hybrids”. Accordingly, in order to determine them one has simply to solve the eigenvalue problem

PASAeiA ) MieiA

(49)

where PA and SA are the intraatomic blocks of matrices P and S, respectively and eAi is the mA-dimensional vector of the atomic natural hybrid.29 (The elements of matrix PA are already determined during the energy calculation, one has only to take care that they are stored explicitly.) Instead of eq 49 containing the non-Hermitian matrix PASA, one may multiply by matrix SA and get

SAPASAeiA ) MiSAeiA

(50)

which is a Hermitian-generalized eigenvalue equation. 4.2. Generalization of the Natural Hybrids. The connection between our LMOs and the concept of “atomic natural orbitals” can be generalized also to cases of other functionals, when the “intraatomic components” of the localized orbitals are not defined in such a simple manner as above (e.g. cases B, C, and E above). For that reason one should consider explicitly b;r b′). The following considerations are valid the expansion of F1(r for any localization functional for which the “intraatomic components” ψAi form an orthogonal set. We known that for a closed-shell determinant wave function any orthonormalized set of occupied orbitals is a natural orbital set with all occupation numbers equal 2. Therefore the firstorder density matrix for the whole determinantal wave function can be written as n

b;r b′) ) 2 ∑ψi(j)*(r b)ψi(j)(r b′) F1(r j)1

(51)

where superscript (j) distinguishes the different solutions ψi obtained for the same functional Fi. The intraatomic part of F1 is obtained if one conserves only the “intraatomic components” ψA(j) of each orbital ψ(j) i i : n

FA1 (r b;r b′)

) 2∑ψiA(j)*(r b)ψiA(j)(r b′)

(52)

j)1

Using eq 48 this transforms to n

b;r b′) ) 2 ∑Mi(j)χiA(j)*(r b)χiA(j)(r b′) FA1 (r

(53)

j)1

indicating that the intraatomic portion of the first-order spinless density matrix is diagonal in terms of the orthonormal set of the effective atomic orbitals χAi introduced above and the occupation numbers equal twice the respective Mi values. The above result offers the following (trivial) generalization of our approach to correlated wave functions: one should diagonalize the intraatomic part of the first-order density matrix in the respective metrics Fi and consider the orbitals obtained as effective atomic orbitals. Of course, it will be more difficult to interpret the results in the correlated case; in particular, comparison with the natural orbitals of free atoms will be inevitable in order to get an insight into the effects related to the purely intraatomic part of the electron correlation. 4.3. Relations to Weinhold’s Analysis. As noted in the Introduction, in the simplest case A of a functional based on Mulliken’s net atomic population, our procedure leads to orbitals somewhat related to, but not identical with, those obtained in the NBO and NAO analysis of Weinhold’s group.1-3 Here we shall discuss this point somewhat more in detail. In the first calculations performed at the INDO level, Foster and Weinhold1 have obtained McWeeny’s natural hybrids but rejected their use because they do not usually have the character of directed hybrids pointed to the neighboring atoms. Instead, as discussed in the Introduction, Foster and Weinhold take from the atomic problem only the core and lone-pair orbitals, while the bonding ones are determined by diagonalizing the different two-center blocks of the first-order density matrix (after the core and lone pairs are separated out). The procedure is then completed by an orthogonalization1 to lead to the set called “natural hybrid orbitals”. The “natural atomic orbitals” (NAOs) of Weinhold and coworkers2,3 are obtained by using the atomic blocks of the density matrix, so are conceptually closer to our approach. In the NAO analysis one first diagonalizes separately the s, p, d, etc. blocks of the atomic density matrix and, by averaging them over the spatial directions (e.g. px, py, pz), gets the so called “pre-NAO”s. The “pre-NAO”s can clearly be assigned either a core, valence or external (“Rydberg”) character. Determining the “pre-NAO”s for all the atoms of the molecule, an orthogonal molecular basis set is constructed by using an ingenious occupation weighted orthogonalization scheme.30 This orthogonal set is called NAOs, and it is formed in such a manner that the strongly occupied NAOs keep much of the similarity with their purely atomic (“pre-NAO”) counterparts. This NAO set is then used to perform the “natural population analysis”, too.2,3 According to the above discussion, the “pre-NAO”s are the closest ones to our “effective AO”s (McWeeny’s hybrids in the case of the simplest localization functional A). However, our orbitals transfrom according to the irreducible representation of the point group of the local symmetry, and usually are not pure (or are only approximately pure) s, p, d, etc. orbitals, reflecting the degree of the asymmetry in the chemical environment of the atom considered. (Especially the presence of lone

Atomic Orbitals from Molecular Wave Functions

a

b

Figure 1. Occupation numbers Mi (in descending order) and approximate characterizations of the effective atomic orbitals in the benzene molecule (DZP basis) obtained by using functional A based on Mulliken’s net populations: (a) carbon and (b) hydrogen atoms.

pairs of σ-type leads to significant s-p mixing, otherwise only some contamination with orbitals of higher angular momentum can mostly be observed.) As noted in the Introduction, we consider an advantage that our procedure leads to effective atomic orbitals for which the orbitals on the same atom are orthogonal but those centered on different atoms are not, because interatomic overlap is very important chemically. We do not consider a serious problem that our orbitals do not usually represent pure s, p, etc. orbitals nor directed hybrids: they permit us to follow how the original atomic orbitals are distorted when the molecule is formed, and one can easily combine them into the proper hybrids if necessary. 5. Examples Figures 1 and 2 display the occupation numbers Mi (in descending order) and approximate characterizations of the relevant effective atomic orbitals for the atoms of the benzene and glycine molecules (DZP and TZP basis sets, respectively) calculated by using functional A. The numeration of the glycine atoms was according to Scheme 1. The number of doubly occupied orbitals is 21 in benzene and 20 in glycine, while the number of basis functions on each non-hydrogenic atom was 15 and 20, and each hydrogen had and 6 and 5 basis orbitals, respectively. Accordingly, there were no serious limitations on the possible number of effective atomic orbitals. Nevertheless, one can see, that there are only five effective AOs having a considerable occupation number Mi on

J. Phys. Chem., Vol. 100, No. 15, 1996 6255 each carbon, nitrogen, and oxygen atom and only one on each hydrogen. This is completely in line with the expectations discussed above. The effective AOs were characterized by inspection of the orbital coefficients; they are either almost pure s or p orbitals, with a slight d orbital admixture, or typical s-p hybrids; the lone pair hybrids have much greater s-character than the σ-bonding ones. (Some of the orbitals denoted pσ have a few percent s-orbital admixture, which is, obviously absent for pπ ones.) The actual Mi values seems quite reasonable from a chemical point of viewse.g., the hydrogen Mi values decrease if the H atom is connected with a more electronegative atom; the oxygen Mi values are greater than those of the carbon, in agreement with the expected direction of the bond polarizations, etc. There are some orbitals (in particular oxygen lone pairs) which have Mi values exceeding one. As Mi characterizes Mulliken’s net population, this is compensated by negative overlap populations related to the “tails” of the orbitals ψi in question, so the normalization condition 〈ψi|ψi〉 ) 1 is fulfilled. (These tails of the lone pairs can be attributed numerous important “directional” effects.) The results are quite similar in other basis sets and other molecules, too, supporting the idea that the electronic structure of many molecules may be very well described in terms of an effective minimal basis on each atom, formed by a set of orbitals which are distorted as compared with the free-atom case but still keep their orthogonality. Figure 3 displays in a condensed form the results of the calculations for the glycine molecule by using the projective functional C and the same TZP basis as in the previous case. As discussed in section 3.3, in this case there is no such sharp limit between the “minimal basis” and the additional orbitals. Nevertheless, the consideration of these data may throw some additional light on the electronic structure prevailing around the different atoms and illustrate how the octet rule is working for them. For the electronegative nitrogen and oxigen atoms there are five (one core and four valence) orbitals which are almost doubly filled (the Mi values are close to unity) indicating that there is a significant contribution of the ionic configurations in the molecular wave function, resulting in an electronic configuration resembling that of neon. This is less the case for the R-carbon C2, but still one can identify the five orbitals of the type mentioned. For the carboxyl carbon C3 the situation is again different, the eigenvalues decrease quite gradually, and the fifth one is as small as 0.631. This behavior is partly connected with the significant positive charge on the carboxyl carbon, and partly with the fact that the overlap of the neighboring pπ orbitals is much lower than that of the oriented σ-hybrids; this small overlap will influence the results for any sp2 carbon atom.31 Figure 4 displays the analogous results for the case when Mulliken’s gross atomic population is used as target function (case D). In this case the distinction of the “minimal basis” is again rather clear, butsas discussed abovesthe atomic orbitals obtained are not strictly orthogonal. One can, however, study in fine detail how the different orbitals contribute to the resulting Mulliken’s gross atomic populations, and clearly see that there are orbitals with negatiVe contributions to the latter: there are some small negative eigenvalues, too. (The populations calculated by using a Lo¨wdin orthogonalized basis or those obtained in Weinhold’s “natural population analysis”2,3 are free of this disadvantage.) 6. Summary In the present paper we have generalized to the case of an arbitrary Hermitian bilinear localization functional our recent

6256 J. Phys. Chem., Vol. 100, No. 15, 1996

Mayer

a

d

b

e

c

f

Figure 2. Occupation numbers Mi (in descending order) and approximate characterizations of the effective atomic orbitals in the glycine molecule (TZP basis) obtained by using functional A based on Mulliken’s net populations: (a) nitrogen, (b, c) carbon, (d, e) oxygen, and (f) hydrogen atoms.

SCHEME 1

idea of extracting effective atomic orbitals from molecular wave function by performing independent localization transformations for each atom separately. The respective general formalism is developed: the localization equations are derived in several different but equivalent forms and are utilized to obtain the orthogonality relationships for the localized molecular orbitals and (for most functionals) their “intraatomic components”, too. The latter can be considered as the effective atomic orbitals

within the molecule. Several different localization functionals are considered and it is shown that for the simplest one the orbitals obtained are natural hybrids in McWeeny’s sense. In this case one obtains for each atom of a “usual” molecule as many effective AOs of appreciable importance as the number of orbitals contained in the classical “minimal basis” of that atom, forming therefore asdistorted but still orthogonalseffective minimal basis of the atom within the molecule. (This is exactly the case in some simple molecules, irrespective of how large the basis is and which localization functional is used.) The similarities and differences with different orbitals of the Weinhold’s analysis are also discussed. It is pointed out that, by selecting a proper localization functional, the present approach

Atomic Orbitals from Molecular Wave Functions

Figure 3. Occupation numbers Mi (in descending order) of the effective atomic orbitals in the glycine molecule (TZP basis) obtained by using functional C based on the projection criterion.

Figure 4. Localization parameters Mi (in descending order) of the effective atomic orbitals in the glycine molecule (TZP basis) obtained by using functional D based on Mulliken’s gross populations.

can also be used to define the effective atomic orbitals in a basis-free manner, i.e. even if no atom-centered basis was used in calculating the wave function. The way to generalize this approach for correlated wave functions is also suggested. Finally, illustrative calculations on benzene and glycine molecules are presented for the case of three different localization functionals. The possibility of extracting effective minimal basis sets of atoms from the molecular wave functions obviously has a great conceptual importance, because it represents a strong corroboration of the classical LCAO picture and indicates the octet rule being inherent in the results of the large scale ab initio calculations. Acknowledgment. The author is indebted to the Hungarian Research Fund for partial financial support (grant OTKA No. T15838) and for providing computing power (grant OTKA No. C0020). References and Notes (1) Foster, J. P.; Weinhold, F. J. Am. Chem. Soc. 1980, 102, 7211.

J. Phys. Chem., Vol. 100, No. 15, 1996 6257 (2) Reed, A. E.; Weinstock, R. B.; Weinhold, F. J. Chem. Phys. 1985, 83, 735. (3) Reed, A. E.; Curtiss, L. A.; Weinhold, F. Chem. ReV. 1988, 88, 899. (4) McWeeny, R. ReV. Mod. Phys. 1960, 32, 335. (5) Mayer, I. Chem. Phys. Lett. 1995, 242, 499. (6) Mayer, I. Can. J. Chem., accepted for publication in the Bader issue. (7) We recall here the well-known fact that any nonsingular transformation among the occupied orbitals leaves the determinant wave function invariantsexcept, possibly, a physically irrelevant constant factor. So, it is sufficient to require all the orbitals considered to lie entirely in the subspace of the occupied one-electron orbitals and be linearly independent. (8) Magnasco, V.; Perico, A. J. Chem. Phys. 1967, 47, 971. (9) Bader, R. F. W.; Nguyen-Dang, T. T. AdV. Quantum Chem. 1981, 14, 63. (10) The author is extremely indebted to Professor Pe´ter Pulay for calling his attention to this fact. (11) We shall recall in this connection that recently somewhat similar “basis free” definitions of the bond order (multiplicity) between two atoms and of the actual valence of an atom in the molecule have also been given. A Ä ngya´n, J. G.; Loos, M.; Mayer, I. J. Phys. Chem. 1994, 98, 5244. (12) We use the subscripts/superscripts i only to remember that each atom in the molecule should be assigned its own functional, its own localized orbitals etc., so for each molecule one has to deal with several similar problems simultaneously. (13) If matrix Fi is positive semidefinite, then this can be done in the subspace spanned by the eigenvectors with nonzero Mi values. (14) Roby, K. R. Mol. Phys. 1974, 27, 81. (15) Pipek, J.; Mezey, P. G. J. Chem. Phys. 1989, 90, 4916. (16) We recall here that Mulliken’s gross atomic population is 2∑µ∈A (PS)µµ; for a single orbital one has P ) cc†. (17) Pipek, J. Int. J. Quantum Chem. 1985, 27, 527; 1989, 36, 487. (18) The term “intraatomic overlap matrix” has a somewhat different meaning for the case of different functionals. In case A it simply gives the overlap of the basis orbitals both belonging to atom A (and is zero otherwise); in case C it represents a “full” matrix formed of the overlap between the projections Pˆ Aχµ of all basis orbitals in the molecule; in cases A as defined by eqs B and D it is the matrix of “atomic overlap integrals” Sµν 27 and 41, respectively. (19) Amos, A. T.; Hall, G. G. Proc. R. Soc. London 1961, 263A, 483. (20) Lo¨wdin, P.-O. J. Appl. Phys. Suppl. 1962, 33, 251. (21) Mayer, I. AdV. Quantum Chem. 1980, 12, 189. (22) For case A it was shown5 that the LMOs ψi can be obtained by solving some mA by mA equations, too (also Vide infra). This may be of importance if one considers extended systems in which the number of occupied orbitals n . mA. (Probably this can also be generalized to case D.) (23) Other variants of the equations may produce some empty effective AOs, too, which we shall not consider here. (24) For these hydrogens one can also obtain up to five effective AOs, similarly to the heavy atoms, but, of course, only one is expected to be of significant importance. (25) Lo¨wdin, P.-O. Phys. ReV. 1955, 97, 1474. (26) Mayer, I. J. Mol. Struct. (Theochem) 1992, 255, 1. (27) A third related mathematical object is matrix SPS consisting of b)|χν〉; also see the integrals of the density operator Fˆ1: (SPS)µν ) 〈χµ|Fˆ1(r refs 2 and 26. (28) Note, however, that one has to consider the intraatomic block of the matrix PSswhich, in general, differs from PASA, and is closely related to the functional D discussed aboveswhen considering problems related to atomic charge and valences (cf. ref 26): for instance it will give AO occupation numbers which (taking them on all atoms) sum up to the number of electrons. (29) Using the notations above, one may write PA ) LiPLi, SA ) LiSLi, eiA ) Liei and one obtains eq 49 from eq 17 by substituting eq 22, multiplying with Li from left, and taking into account that LiLi ≡ Li. (30) The directional averaging forces the “pre-NAO”s to keep the spherical symmetry of the free atoms, but in an unsymmetrical environment this symmetry is necessarily spoiled out to some extent by the orthogonalization when the final NAOs are formed. (31) An unpolarized two-center bonding orbital formed of the AOs with overlap S, has a projection [2(1 + S)]1/2/2 on each of the atoms.

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