ORBITALS AND SLATER DETERMINANTS

We begin our discussion of wave function–based quantum chemistry by introducing the concepts of n-electron and one-electron expansions. First, in Sec. 2.1, we consider the expansion of the approximate wave function in Slater determinants of spin orbitals. Next, we introduce in Sec. 2.2 the one-electron Gaussian func-tions (basis funcfunc-tions) in terms of which the molecular spin orbitals are usually con-structed; the standard basis sets of Gaussian functions are finally briefly reviewed in Sec. 2.3.

2.1. Slater Determinants andn-Electron Expansions

The construction of approximate electronic wave functions is a difficult many-body problem. The source of the difficulties is the presence of the two-body electron–elec-tron repulsion term in the Hamiltonian equation [Eq. (3)]. In the absence of this term, there would be no interactions among the electrons and it would be sufficient to consider one electron at the time, independently of the others. Indeed, in this case, the

many-electron Hamiltonian equation [Eq. (3)] becomes, apart from a constant nuclear–nuclear repulsion term, a sum of independent one-electron Hamiltonians:

ˆH^nonint¼X

are called orbitals. To account for the two observed spin states of the electron, we introduce spin orbitals as products of such orbitals with one of two possible spin states:

u_kað Þuux ka where the spin coordinate s takes on the values 1/2 and
1/2. The spin functions are given by:

For brevity of notation, we shall in the following include the spin part of the spin orbitals in the spin orbital label:

u_jð Þ ¼ ux kj

!r

ð Þrjð Þ;s ð11Þ

where rj(s) is either a(s) or b(s), depending on the spin orbital label j. For a given set of orbitals, there are twice as many spin orbitals.

A many-electron wave function may now be written as a product of these spin orbitals, properly antisymmetrized to comply with the Pauli principle. For two electrons, for example, we obtain:

A_k;lðx₁; x2Þ ¼ 1ffiffiffi

p u2½ kð Þux₁ lð Þ
ux₂ kð Þux₂ lð Þx₁ ; ð12Þ which represents a state of noninteracting electrons. Note that this state vanishes when the two spin orbitals are identical or when x1=x2. More generally, for a system of n independent electrons, the wave function may be written as a Slater determinant; that is, as the determinant of a square matrix whose elements are spin orbitals, with the electron labels as row indices and spin orbital labels as column indices:

A_lðx₁; x2; . . . ; xnÞ ¼ 1ffiffiffiffi

sometimes abbreviated as Al=jul₁,ul₂,. . . ,ul_nj. In passing, we note that, in the following, we shall often use the word ‘‘configuration’’ for ‘‘Slater determinant.’’

The term ‘‘configuration’’ is short for ‘‘configuration state function,’’ which is a spin-symmetrized and space-spin-symmetrized linear combination of Slater determinants.

A Slater determinant gives an exact representation of the n-electron wave function only in the (fictitious) limit of no interactions among the electrons (i.e., for a system of electrons described by the Hamiltonian in Eq. (4)). For a real system of interacting electrons, described by the Hamiltonian in Eq. (3), the Slater determinant can only serve as an approximate wave function. Nevertheless, in this case, we may still represent the true n-electron wave function C exactly as a linear combination of Slater determinants (Eq. (13)):

C xð ₁; x2; . . . ; xnÞ ¼X^Ndet

l¼1

clA_lðx₁; x2; . . . ; xnÞ; ð14Þ

where Ndet is the number of unique n-electron Slater determinants that may be constructed from N spin orbitals:

Ndet¼ N n



: ð15Þ

This representation of the wave function is exact provided that a complete set of exact eigenfunctions (spin orbitals) of hˆiin Eq. (5) is used to construct the Slater determi-nants. In practical calculations, we do not have access to a complete set of orbitals.

Moreover, even if we had these orbitals, we would not be able to construct the full set of Slater determinants that they give rise to, noting that the number of determinants (Eq. (15)) increases very steeply with the number of orbitals.

In practice, therefore, we must work with finite-dimensional orbital spaces, properly optimized so as to yield the best representation of the n-electron wave func-tion. In addition, we must, for a given finite orbital basis, find a way to determine the coefficients in the expansion (Eq. (14)). The ‘‘best’’ wave function (i.e., the wave function with the lowest energy) is obtained by optimizing all expansion coeffi-cients clvariationally, as done in the full configuration interaction (FCI) method [18–

20]:

However, this approach is too expensive because, even for very small molecules such as HF in moderately small orbital spaces, Ndetbecomes very large—several billions or more. Thus, we must instead be content to work with approximate FCI wave functions. Fortunately, several useful hierarchies of approximations to the FCI solution have been developed, enabling us to approach the FCI wave function closely, even for rather large systems.

From our discussion, it should be clear that the quality of our approximate solution to the Schro¨dinger equation depends not only on the particular n-electron model by which we choose to approximate the FCI solution, but also on the set of orbitals from which the FCI solution or its approximations are constructed. Before we consider in Secs. 3 and 4 the various techniques that have been developed to

ap-proximate the FCI wave function (Eq. (14)), we shall in the remainder of this section focus our attention on the spin orbitals.

2.2. Atomic Orbitals

In systems of high symmetry such as atoms, it is possible to represent the orbitals (i.e., the spatial part of the spin orbitals) numerically on a spatial grid. For polyatomic systems, however, it is more common to represent the spin orbitals as linear expansions of a set of N simple, analytical one-electron basis functions vj(x), mostly centered on the atoms in the system. These linear expansions may then be written as:

u_kð Þ ¼x X

j

Cjkv_jð Þ;x ð17Þ

where, as before, the spin part has been included in the basis functions:

v_jð Þ ¼ vx _kjð Þr^!r jð Þ:s ð18Þ

In the following, the functions vj(x) with 1 V j V 2N are understood to be spin-dependent, whereas the functions v_kð^!rÞ with 1 V j V N are pure spatial functions.

The nonorthogonal basis functions v_kð^!rÞare referred to as atomic orbitals (AOs) and are often taken to be Cartesian Gaussian-type orbitals (GTOs) of the (unnormal-ized) form:

Gijk !r; a; ^!A

 

¼ xⁱ_Ay_A^jz_A^kexp
ar_A²

; ð19Þ

where a>0 is the orbital exponent and where:

rA¼ j^!r
^!Aj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²_Aþ y²_Aþ z²_A

q : ð20Þ

The nonnegative integers i, j, and k in Eq. (19) are related to the ‘‘angular momentum’’

of an electron in this AO as l = i+j+k. Gaussian functions are nearly always added in full shells (i.e., for a given orbital exponent a and a given l, all components i+j+k=l are included in the basis simultaneously, thereby treating all Cartesian directions equivalently.

In the Cartesian scheme (Eq. (19)), there are (l+1)(l+2)/2 components of a given l, whereas the number of independent spherical harmonics is only 2l+1. Usually, therefore, the Cartesian GTOs are not used individually but instead are combined linearly to give real solid harmonics (see Ref. 1). In addition, for a more compact and accurate description of the electronic structure, the GTOs (Eq. (19)) are not used individually as primitive GTOs but mostly as contracted GTOs (i.e., as fixed, linear combinations of primitive GTOs with different exponents a).

Although most molecular calculations are carried out using GTOs, in some cases (in particular for atoms and diatoms), Slater-type orbitals (STOs) are used instead. The STOs have a different radial form than the GTOs, proportional to exp(
frA) rather than to exp(
ar²_A). The GTOs are used in preference to the STOs because the evaluation of many-center integrals is much easier for GTOs than for STOs.

2.3. Gaussian Basis Sets

Over the years, a variety of standard Gaussian basis sets have been developed for virtually all atoms of the periodic table [21,22]. For qualitative or exploratory work, minimal basis sets, which contain only one shell of AOs for each (fully or partially)

occupied shell in the parent atom, may be used. A popular minimal basis set is the STO-3G basis, where each AO is a linear combination of three primitive GTOs.

For semiquantitative work, double-zeta or triple-zeta basis sets (in which there are two or three shells of AOs for each fully or partially occupied atomic shell) are needed, at least for the valence shell. For first-row atoms, for example, the popular 6-31G basis has a minimal representation of the 1s core orbital and a double-zeta representation of the valence orbitals. Moreover, each AO is represented by a fixed linear combination of primitive GTOs: the 1s core orbital contains six primitive AOs, and each 2s and 2p valence orbital is represented by two contracted orbitals, containing three and one primitive functions.

In addition, polarization functions are needed. Such functions, which are of dif-ferent symmetry than the AOs in the parent atom, are needed to describe the po-larization of the atomic charge in the molecular environment. In the 6-31G* basis, for example, the 6-31G basis is augmented with a set of d-type polarization functions on the first-row atoms; in the 6-31G** basis, polarization functions ( p-type) are also added to the hydrogens.

The basis sets described above are small and intended for qualitative or semi-quantitative, rather than semi-quantitative, work. They are used mostly for simple wave functions consisting of one or a few Slater determinants such as the Hartree–Fock wave function, as discussed in Sec. 3. For the more advanced wave functions discussed in Sec. 4, it has been proven important to introduce hierarchies of basis sets. New AOs are introduced in a systematic manner, generating not only more accurate Hartree–

Fock orbitals but also a suitable orbital space for including more and more Slater determinants in the n-electron expansion. In terms of these basis sets, determinant expansions (Eq. (14)) that systematically approach the exact wave function can be constructed . The atomic natural orbital (ANO) basis sets of Almlo¨f and Taylor [23]

were among the first examples of such systematic sequences of basis sets. The ANO sets have later been modified and extended by Widmark et al. [24].

The correlation-consistent basis sets of Dunning [25], Kendall et al. [26], and Woon and Dunning [27] provide a particularly popular hierarchy of basis sets, which has been extensively used to extrapolate toward the FCI limit of a complete AO basis.

For calculations correlating only the valence electrons, these basis sets are denoted cc-pVXZ, where 2 V X V 6 is the cardinal number [25]. For first-row atoms, the smallest double-zeta basis cc-pVDZ with X = 2 contains three s-type contracted GTOs, two sets of p-type contracted GTOs, and one set of d-type GTOs (in total, 14 contracted AOs). At the next level, the triple-zeta cc-pVTZ basis with X = 3 contains 30 AOs, followed by the quadruple-zeta cc-pVQZ basis with 55 AOs, and the cc-pV5Z basis with 91 AOs. For first-row atoms, the number of AOs in the cc-pVXZ basis is given by 1/3(X+1)(X+3/2)(X+2). The largest correlation-consistent basis, cc-pV6Z, repre-sents a 7s6p5d4f 3g2h1i basis of 140 contracted GTOs. We note, however, that the cc-pVXZ basis sets are constructed for correlating only the valence electrons. For correlating all electrons (core as well as valence electrons), the correlation-consistent core valence basis sets cc-pCVXZ with 2 V X V 5 are used [27].