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Keywords:

  • effective atomic orbitals;
  • effective minimal basis;
  • Hilbert-space analysis;
  • 3D analysis

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. AOs from Molecular Wave Functions
  5. Some Examples
  6. Interrelations between Hilbert-Space and 3D Analyses
  7. Conclusions and Perspectives
  8. Acknowledgment

It is discussed that one can obtain effective atomic orbitals (AOs) in quite different theoretical frameworks of Hilbert-space and three-dimensional (3D) analyses. In all cases, one can clearly distinguish between the orbitals of an effective minimal basis set and orbitals which are only insignificantly occupied. This observation makes a solid theoretical basis beyond our qualitative picture of molecular electronic structure, described in terms of minimal basis AOs having decisive participation in bonding, and may be considered as a quantum chemical manifestation of the octet rule. For strongly positive atoms like the hypervalent sulfur, some weakly occupied orbitals reflecting “back donation” can also be identified. From the conceptual point of view, it is very important that AOs of characteristic shape can be obtained even by processing the results of plane wave calculations in which no atom-centered basis orbitals were applied. The different types of analyses (Hilbert-space and 3D) can be done on equal footing, performing quite analogous procedures, and they exhibit an unexpected interrelation, too. © 2014 Wiley Periodicals, Inc.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. AOs from Molecular Wave Functions
  5. Some Examples
  6. Interrelations between Hilbert-Space and 3D Analyses
  7. Conclusions and Perspectives
  8. Acknowledgment

Atomic orbitals (AOs) represent an important concept in the theory of atomic and molecular physics. Historically, they first appeared as exact solutions of the Schrödinger equation of the free hydrogen atom, then they have been obtained as solutions in the Hartree–Fock (HF) approximation for the free many-electron atoms. The HF “canonical” orbitals are qualitatively similar to the hydrogenic ones, but without the “accidental” degeneracy of the s, p, and so forth orbitals of the same principal quantum number. Accounting for some peculiarities in the order of the orbital energies and the corresponding shell-filling scheme (such as 3d vs. 4s for transition metals), the use of the AO concept permits one to rationalize the periodic system of the elements and to specify different electronic states of the free atoms.

Started with the classical work of Heitler and London, the first numerical calculations on molecular systems (basically H2 in the very first period) were performed using the orbitals of free atoms (maybe squeezed and/or polarized) as the indispensable tools. Then, the “founding Fathers” of quantum chemistry (who had not the possibility to perform large-scale calculations but had the ability of deep thinking) introduced a number of qualitative/semiquantitative concepts, such as linear combination of atomic orbitals (LCAO), hybridization, delocalization, and so forth, until now forming the basis for our qualitative interpretation of electronic structure of molecules—always in terms of different AOs. The first ab initio calculations (and, in fact, also the semiempirical ones) have been performed in terms of the “minimal basis” of AOs, either those of the free atoms, or using some approximation to them in terms of Slater-type or (contracted) Gaussian-type “basis orbitals.”

With the development of the computers and computational techniques, these concepts basically disappeared from the quantum chemical discourse, much more oriented to the numbers—energies, geometrical parameters, and so forth. LCAO has “survived” in the sense that most calculations use atom-centered basis sets, although in the last years more and more calculations are performed by applying plane wave basis sets, that is, without using explicitly the LCAO concept. By the use of larger and larger basis sets, the concept of well-defined AOs making up the molecular orbitals does not appear in a routine quantum chemical calculation—although it is probably crucial for a proper understanding of the results of these calculations. For instance, to interpret planarity of the benzene molecule, one can hardly avoid the concept of sp2 hybridization of the carbon's AOs.

In such a situation, it represents a conceptual interest how the classical picture of minimal basis core and valence orbitals can be recovered by an appropriate a posteriori analysis of wave functions—the present article is devoted to this problem. For sake of simplicity, we shall concentrate on closed shell systems treated at the single determinant (HF or DFT) level of theory.

As known, there are two main approaches for doing a posteriori analysis of wave functions: the Hilbert-space analysis and the three-dimensional (3D) analysis.[1] In the former case, one considers the atom in molecule as an entity defined by the nucleus and the basis orbitals assigned to it, whereas in the latter one the atom is defined as the nucleus and a disjunct or “fuzzy” domain of the 3D physical space around it. In our analysis, we shall pursue both types of analysis simultaneously, using a common formalism, and then, we shall discuss some interrelations between these two approaches.

Of course, there were several approaches to the problem of AOs within a molecule during the decades, using different theoretical approaches and introducing different criteria to determine the AOs. We should first of all mention the “effective hybrids” of McWeeny,[2] diagonalizing the intraatomic block of the density matrix (in an orthonormalized basis), our approach may be considered a direct generalization of his effective hybrids. The “modified atomic orbitals” of Heinzmann and Ahlrichs[3] were based on the concept of Roby's charge,[4] and minimized the “unassigned charge,” not accounted for by the minimal basis modified AOs. In the natural atomic orbital (NAO) analysis of Weinhold and coworkers,[5, 6] a rather complex recipe is used to obtain a set of NAOs which can be assigned core, valence, or “Rydberg” character and are also used to perform the natural population analysis. While one often can get useful chemical insight using these NAOs, we feel lacking a clear mathematical criterion (some target functional) behind them. Also, as the final NAOs of a whole molecule are obtained orthonormalized, they do not carry any direct information about the (chemically very important, in our opinion) effects of interatomic overlap. In[7], we have proposed to use the Magnasco–Perico localization criterion[8] (maximization of Mulliken's net atomic population) to determine the molecular orbitals (MOs) from which effective AOs can be obtained; then, the approach was generalized to an arbitrary Hermitian bilinear localization functional[9]; actually the use of Mulliken's gross populations was also tested numerically (and rejected). The approach has been generalized to the 3D analysis, too.[10-13] The “free” (or broken) valences obtained in the framework of the domain-averaged Fermi hole (DAFH) formalism of Ponec et al.[14, 15] are entities very close to the AOs forming chemical bonds; despite the fact that the DAFH formalism is an explicitly two-electron scheme and uses the second-order density matrix, in the single determinant (HF or Kohn–Sham) case, it reduces to the calculation of group functions exactly in the manner of our theory,[16, 17] showing that these approaches are closer to each other than it would appear at the first sight.

AOs from Molecular Wave Functions

  1. Top of page
  2. Abstract
  3. Introduction
  4. AOs from Molecular Wave Functions
  5. Some Examples
  6. Interrelations between Hilbert-Space and 3D Analyses
  7. Conclusions and Perspectives
  8. Acknowledgment

The solutions of the HF or Kohn-Sham equations are delocalized canonical MOs. However, exactly the same wave function can be described also by different sets of localized MOs. We shall use a special localization scheme permitting to restore the AOs actually participating in building the MOs.

Atomic parts of the MOs

At first, we define the MOs as representing sums of some “atomic parts”

  • display math(1)

The definition of the atomic part inline image of the MO inline image depends on whether one uses Hilbert-space or 3D analysis.

In the Hilbert-space analysis, one starts from the LCAO expansion of the MO

  • display math(2)

and only the basis orbitals centered on atom A are conserved in the expansion to form the atomic part inline image of the MO inline image:

  • display math(3)

In the case of the Bader's “atoms in molecules” (AIM) analysis—or, in general, if the 3D physical space is decomposed into the disjoint atomic domains inline image—the definition of the atomic part of an orbital inline image is simply

  • display math(4)

In the “fuzzy atoms” analysis, there are no sharp boundaries of the atoms. In this case, we define in every point inline image of the space the atomic weight functions inline image that are large inside the respective atom A, small outside it, and satisfy

  • display math(5)

Then, the atomic part of the orbital inline image is defined as

  • display math(6)

The localization procedure and the effective AOs

We perform a separate localization for each atom, by requiring

  • display math(7)

that is, the atomic part inline image of each localized orbital inline image must have a maximal (at least stationary) norm.

By writing the localized orbital as a linear combination of the canonical ones:

  • display math(8)

where n is the number of the (doubly filled) occupied orbitals, the stationarity requirement, in a standard manner, leads to the eigenvalue equation

  • display math(9)

Here, the Hermitian matrix inline image has the elements

  • display math(10)

where inline image is the atomic part of the (normalized) canonical orbital inline image, defined in the manner discussed above, and inline image is the diagonal matrix of the eigenvalues

  • display math(11)

When doing this derivation, we have taken into account that forming the atomic part of an orbital is a linear procedure in each case discussed above, so expansion (8) holds for the atomic parts of the respective orbitals too and one can write

  • display math(12)

These equations were obtained for the different schemes of analysis[7-11] independently from each other, although a general mathematical formalism has also been discussed.[9] As can be seen here, one should solve quite similar equations independently of the type of analysis. Also, it is common that the atomic parts of the localized orbitals after renormalization define an orthonormalized set of effective AOs inline image with the eigenvalues Mi being their occupation numbers (net atomic populations of the respective localized orbitals). This means that

  • display math(13)

Orbitals inline image are indeed orthonormalized:

  • display math(14)

owing to the fact that the unitary matrix inline image diagonalizes matrix inline image.

One can see by inspection that orbitals inline image represent the results that one obtains by performing Löwdin's canonic orthogonalization[18] of the atomic components inline image of the canonical orbitals inline image.

In the case of a Hilbert-space analysis, the number of the effective AOs of the given atom, having nonzero occupation numbers, is limited by the smaller of the number of the basis orbitals of that atom and the number of MOs in the molecule. In the 3D formalism, one gets, in general, as many effective AOs for each atom, as is the number of MOs in the molecule.

The effective AOs as natural hybrids

It is known that the first-order spinless density matrix can be expressed by the natural orbitals inline image and their occupation numbers ni as

  • display math(15)

One can define the (intra)atomic part of the spinless first-order density matrix in terms of the atomic parts inline image of the natural orbitals, formed according to the schemes discussed above:

  • display math(16)

In the case of a single closed shell determinant wave function, all the occupation numbers ni are equal either 2 or 0, and Eq. (16) becomes

  • display math(17)

In the closed-shell single determinant case, one has the unitary invariance property; it is easy to check that it also holds to Eq. (17). That means that any set of orthonormalized occupied orbitals may equally well be used in these expressions, including those in Eq. (12), obtained by solving the eigenvalue equation of matrix inline image for the given atom:

  • display math(18)

It is known that the first-order spinless density matrix inline image is (up to a factor of inline image) the kernel of the projection operator inline image onto the subspace of the occupied orbitals, and the eigenvectors of the latter are the natural orbitals; using the “bra-ket” notations that can be written as

  • display math(19)

Equation (13) can be written as inline image using which Eq. (18) can be rewritten as

  • display math(20)

This can also be considered a kernel of an integral operator, and one can write, in analogy to Eq. (19)

  • display math(21)

where the orthonormalization property Eq. (14) has been utilized. This result indicates that the effective AOs inline image may be considered direct generalizations of the “natural hybrid orbitals” introduced by McWeeny,[2] with the occupation numbers inline image representing their net atomic populations.

In the case of a correlated wave function, the present approach can easily be generalized by considering the atomic part of the spinless first-order density matrix, defined in Eq. (16), as a kernel of an integral operator and solving the eigenvalue equation

  • display math(22)

giving the effective AOs in the correlated case.

The effective minimal basis

Experience shows that for nonhypervalent systems there are always as many effective AOs with Mi values significantly greater than zero, as is the number of AOs in the classical minimal basis of the atom. They define an effective minimal basis of the atom. This is the case at any types of treatment—Hilbert-space,[7, 9] fuzzy-atoms[11] as well as for Bader's AIM[13]—for all “normal valence” compounds, while for hypervalent compounds a small “shoulder” is observed on the curves of occupation numbers, reflecting the effects of “back donation” to (usually d-type) orbitals. It is important to stress that this observation is not connected with the use of any atom-centered basis set, but is valid even for the pure plane wave calculations,[12] so it indeed reflects some general property of molecular wave functions.

This observation may be considered as a quantum chemical manifestation of the octet rule: for normal valence nonhydrogenic atoms, one has (besides the core shells) four orbitals in a classical minimal basis. The observation mentioned means that the number of MOs with a considerable contribution from the valence-type basis orbitals of the given atom is also four; in a closed-shell molecule, these four orbitals carry eight electrons in accord with the octet rule. (The back-donation effects for hypervalent atoms do somewhat modulate but not invalidate this conclusion.)

Some Examples

  1. Top of page
  2. Abstract
  3. Introduction
  4. AOs from Molecular Wave Functions
  5. Some Examples
  6. Interrelations between Hilbert-Space and 3D Analyses
  7. Conclusions and Perspectives
  8. Acknowledgment

During the years, we have considered the effective AOs in different frameworks. First, we have obtained effective AOs in classical Hilbert-space analysis,[7, 9] for which a free program has also made available,[19] then for the “fuzzy atoms” analysis[11] of standard wave functions built up from atom-centered basis sets. Most recent are the effective AOs obtained in “fuzzy atoms” analysis accomplished for calculations using plane wave basis,[12] as well as the calculations in the framework of the Bader's AIM analysis.[13]

Here, we are going to consider only a few examples. Figures 1 and 2 show typical results obtained in the framework of the Hilbert-space analysis, accomplished with the program[19] mentioned. These figures display the occupation number calculated for the different atoms of the glycine and CH3SO2 molecules, respectively, both using the cc-pVTZ basis set. (According to our experience, this basis is very well suited for a posteriori analyses of the quantum chemical results, as it is well balanced and combines flexibility with a pronounced atomic character of the basis functions.)

image

Figure 1. Occupation numbers of the effective AOs in glycine molecule (cc-pVTZ).

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image

Figure 2. Occupation numbers of the effective AOs in the CH3SO2Cl molecule (cc-pVTZ).

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It can be seen that all the hydrogen atoms exhibit only a single orbital with nonnegligible occupation number, in accord with the fact that hydrogens have only one valence orbital. The carbon, nitrogen, and oxygen atoms have five effective AOs with significant occupation numbers—again in full agreement with the classical chemical notions: one 1s core and four 2s−2p valence orbitals. Similarly to that, the “normal valence” chlorine atom in the CH3SO2 molecule has nine effective AOs: there is an additional completely filled 2s−2p shell as compared with C, N, or O, so the valence orbitals belong to the 3s−3p shell.

However, the hypervalent sulfur, that has a formal valence equal six, exhibits four additional effective AOs with small but by far not negligible occupation numbers; they practically consist of d-type atomic basis orbitals only. Considering hypervalency, it is worth mentioning that the occupation numbers displayed on the figure represent the net orbital occupations of the effective AOs; although these are small, the contribution of these orbitals to the total valence is quite considerable and the resulting valence of 5.816 is approaching the formal value of 6, see Table 1.

Table 1. Partial valences of the different atoms in the CH3SO2Cl molecule calculated by using cc-pVTZ basis set.
AtomValence contribution
spdfTotal
C0.8872.8880.0680.0043.847
H0.9320.0520.0040.987
H0.9360.0550.0040.996
H0.9320.0520.0040.987
S0.8802.9611.7770.1995.816
O0.2641.7760.0290.0022.070
O0.2641.7760.0290.0022.070
Cl0.0400.9610.0460.0041.051

Figure 3 shows the four strongly occupied valence orbitals and the first four weakly occupied (d-type) orbitals of the sulfur atom in the SF6 molecule, extracted by the “fuzzy atoms” analysis of the plane wave DFT calculation.[12] The orbitals that were obtained without using any atom centered basis set resembles completely the classical sp valence basis, if one takes into account that their outer parts are “cut” in accordance with the quickly decreasing atomic weight function inline image. The emergence of the atomic d-orbitals from the results of plane wave calculations is also quite reassuring.

image

Figure 3. The strongly occupied valence and weakly occupied d-orbitals of the sulfur atom in the SF6 molecule, extracted from plane wave results.

(Reproduced with permission from I. Mayer, et al., J. Phys. Chem. A 2011, 115, 12733. © American Chemical Society.)

[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Figure 4 shows the effective AOs of the carbon atom in methane molecule and their occupation numbers, obtained from a Bader-analysis of a B3LYP/cc-pVTZ calculation. Again, the orbitals essentially look as usual, only they are strictly limited to the AIM atomic domain with sharp boundaries, as they should.

image

Figure 4. Effective AOs of the carbon atom in methane molecule and their occupation numbers: Bader-analysis.

(Reproduced with permission from E. Ramos-Cordoba, et al., J. Chem. Phys. 2013, 138, 214107. © American Institute of Physics.)

[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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From our point of view, we may call most appropriate those basis sets and 3D atomic definitions which show the sharpest drop of the occupation numbers Mi after the functions of the minimal basis orbitals for conventional organic compounds. For such basis sets (atom definitions), the appearance of shoulders in the occupation numbers (as that discussed above for the hypervalent sulfur) may be considered significant from the physicochemical point of view.

Interrelations between Hilbert-Space and 3D Analyses

  1. Top of page
  2. Abstract
  3. Introduction
  4. AOs from Molecular Wave Functions
  5. Some Examples
  6. Interrelations between Hilbert-Space and 3D Analyses
  7. Conclusions and Perspectives
  8. Acknowledgment

The Hilbert-space and the 3D-type analyses are much different conceptually, and exhibit quite different behavior when the basis set is changed. The results of the Hilbert-space analysis is subjected to a strong basis-dependence, due to which one should compare only results obtained with the use of strictly the same basis set. (The large sensitivity of the Mulliken-populations to the basis applied is the best known example of this problem.) Also, the Hilbert-space results related to the individual atoms (pairs of atoms etc.) are lacking any complete basis-set limit: in principle, one can use a complete basis set located anywhere in the space, so the concept of atomic basis functions loses its meaning as the basis set becomes complete. The well-known consequence of these limitations is also that no meaningful results can be expected from a Hilbert-space analysis if the basis set contains diffuse functions that lack any pronounced atomic character.

From other side, the 3D results usually exhibit quite smooth convergence to some limiting values as the basis sets improve; they are, however, rather sensitive to the manner in which the (disjunct or fuzzy) atomic domains are defined, that is to the actual values of the individual weight functions inline image in different points of the space.

The possibility to develop the analysis using a common formalism, as it has also been done above, as well as the existence of some rules of “mapping” between the two schemes of analysis[22] stressed their common features. Nonetheless, in light of the sharp difference between them, it has been somewhat surprising to get the recent results[13, 23] permitting to find a specific interrelation between the 3D analysis and a specific case of the Hilbert-space analysis. The point is that the analysis described in the section AOs from Molecular Wave Functions gives in the 3D case, in general, as many effective AOs for each atom as is the number of the MOs in the system. The first AOs among them are the orbitals of the effective minimal basis, while all together span the same subspace as the “atomic parts” inline image of the original (canonical) MOs inline image. Therefore, the effective AOs of all the atoms together form a basis, in which the molecular wave function can be expanded. As each of the orbitals of that basis belongs to a well-defined atom in the system, one can perform a conventional Hilbert-space analysis using this special basis set.

Now, it appears that for that special basis set the Hilbert-space analysis gives exactly the same results as the 3D one: Mulliken's net (qAA), overlap (qAB), and gross (QA) populations are strictly equal to their 3D counterparts and the same holds for the bond orders (BAB),and, therefore, for the valences, too:

  • display math(23)

where the abbreviation LCEAO means “linear combination of effective atomic orbitals.” The proof of equalities (23) is simple but little involved, so, here, we refer only to the original publications.[13, 23]

This result indicates that the known problems with the Mulliken-type analyses are not related with the formalism, but rather to the basis set used: a basis which is good for energetic calculations is often not adequate for doing qualitative LCAO analyses, because contains a number of practically empty orbitals having significant overlaps with the occupied ones.

Conclusions and Perspectives

  1. Top of page
  2. Abstract
  3. Introduction
  4. AOs from Molecular Wave Functions
  5. Some Examples
  6. Interrelations between Hilbert-Space and 3D Analyses
  7. Conclusions and Perspectives
  8. Acknowledgment

We have discussed that one can obtain effective AOs (generalized natural atomic hybrids) in quite different frameworks—Hilbert-space analysis, Bader-type, or “fuzzy atoms” 3D analysis of the results of conventional quantum chemical calculations, as well as “fuzzy atoms” 3D analysis of those of the plane wave calculations. In all cases, one can clearly distinguish between the orbitals of an effective minimal basis set and orbitals which are only insignificantly occupied. Although these orbitals do not strictly coincide with the free atomic ones, this observation makes a solid theoretical basis beyond our qualitative picture of molecular electronic structure, described in terms of minimal basis AOs having decisive participation in bonding, and may be considered as a quantum chemical manifestation of the octet rule. For strongly positive atoms like the hypervalent sulfur, some weakly occupied orbitals reflecting “back donation” can also be identified. (For sulfur they are of d-type, as expected.) From the conceptual point of view, it is very important that AOs of characteristic shape can be obtained even by processing the results of plane wave calculations in which no atom-centered basis orbitals were applied.

The different types of analyses (Hilbert-space and 3D) can be done on equal footing, performing quite analogous procedures, and they exhibit an unexpected interrelation: the Hilbert-space analysis performed in the basis of the effective AOs obtained in the 3D analysis gives results coinciding with those obtained directly in the 3D formalism.

Until now, our attention has mainly been focused on obtaining the set of effective AOs in such a manner that the orbitals forming the effective minimal basis is distinguished from the other ones as sharply (as far as occupation numbers are concerned) as only possible. It has been observed that the minimal basis orbitals are resembling pure s and p ones for symmetric molecules and often represent some hybrids in the general case. It would be worth to study the detailed spatial form and directionality of the different effective minimal basis orbitals to be able to discuss the different steric effects, and perhaps relations to the VSEPR model too.

Another perspective approach may be the investigation of the energetic effects which are connected with the distinction between the full set of effective AOs and the effective minimal basis. One may, for instance, study how large energetic change takes place if one omits all—or some—of the weakly occupied orbitals from the basis of effective AOs. Alternatively, it may be of interest to calculate the (energetically) best minimal basis by direct variational calculation, instead of calculating the effective minimal basis orbitals from an a posteriori analysis of the results of an already accomplished calculation.

Acknowledgment

  1. Top of page
  2. Abstract
  3. Introduction
  4. AOs from Molecular Wave Functions
  5. Some Examples
  6. Interrelations between Hilbert-Space and 3D Analyses
  7. Conclusions and Perspectives
  8. Acknowledgment

The author is indebted to his coauthors of different relevant papers, in particular to Dr. Pedro Salvador, Dr. Imre Bakó, and Dr. András Stirling for the fruitful common work and to the Hungarian Research Fund (OTKA) for the continuous financial support of his research during the last decades.

  1. 1

    In the plane-wave calculations, one often uses pseudopotentials to replace inner shell electrons. That is, of course, in some sense also an application the LCAO concept.

  2. 2

    Of course, one might distinguish the sets localized orbitals inline image corresponding to different atoms by introducing an additional subscript/superscript, but that would lead to very cumbersome notations.

  3. 3

    Not to be confused with Löwdin's more common “symmetric” orthogonalization.

  4. 4

    On these graphs, the quantity inline image is twice of that discussed above.

  5. 5

    The partial valences displayed in this table have been calculated as sums of the bond orders[20, 21] formed by the atomic basis orbitals of the given type. The results are in agreement with the chemical expectations: hydrogen forms one bond by its one s-type orbital, for carbon contribution of each of the 2s and 2p orbitals is nearly one; the bonds of the oxygen and fluorine are formed mainly by their p-orbitals. The [rather positive] hypervalent sulfur forms four bonds like carbon, and the additional two are due to the—weakly populated but still present—d-orbitals resulted from “back-donation.”

  6. 6

    Moreover, one may state that the fact that the 3D populations coincide with the Mulliken populations calculated in the basis of the effective AOs, supports once again our point of view first expressed more than three decades ago,[20] according to which the “halving” of the overlap populations characteristic for the Mulliken analysis is not an arbitrary choice and, therefore, Mulliken-populations have privileged mathematical importance in the framework of the LCAO formalism.

  7. 7

    As a Referee has stressed, the analyses of such type will be worth to be extended to systems like transition metals in typical bonding situations, and to check whether the results correspond to the appropriate generalizations of the octet rule—as “18 electron rule”—applied for such systems.

  8. 8

    A connection between hybridization and the VSEPR rules has already been discussed in.[24]