SCF Wave Functions for Diatomic Molecules

This section presents some examples of SCF wave functions for diatomic molecules. (For a summary of SCF calculations on diatomic molecules, see MulUken and Ermler, Diatomic Molecules.) 

SCF wave functions using a minimal basis set were calculated by Ransil for several light diatomic molecules [B. J. Ransil, Rev. Mod. Phys., 32,245 (I960)]. As an example, the SCF MOs for the ground state of U2 [MO configuration (l rg) l(r ) 2xrg) at R = Rf are 

The AO functions in these equations are STOs, except for 2sj. A Slater-type 2s AO has no radial nodes and is not orthogonal to a Is STO. The Hartree-Fock 2s AO has one radial node (n — / — 1 = 1) and is orthogonal to the Is AO. We can form an orthogonal-ized 2s orbital with the proper number of nodes by taking the following normalized linear combination of Is and 2s STOs of the same atom (Schmidt orthogonalization)  

Since six AOs were used as basis functions, the Roothaan equations yielded approximations for the six lowest MOs of ground-state U2 only three of these MOs are occupied. The expressions for the other three can be found in Ransil s paper. 

Comparison of these with (13.182) shows the simple LCAO functions to be reasonable first approximations to the minimal-basis-set SCF MOs. TTie approximation is best for the la-g and MOs, whereas the 2a-g MO has substantial Iptr AO contributions in addition to the 2s AO contributions. For this reason the notation of the third column of Table 13.1 (Section 13.7) is preferable to the separated-atoms MO notation. The substantial amount of 2s-2pa- hybridization is to be expected, since the 2s and 2p AOs are close in energy [see Eq. (9.27)] the hybridization allows for the polarization of the 2s AOs in forming the molecule. 

This section presents some examples of SCF MO wave functions for diatomic molecules. 

The spatial orbitals in an MO wave function are each expressed as a linear combination of a set of one-electron basis functions Xs- 

For SCF calculations on diatomic molecules, one can use Slater-type orbitals [Eq. (11.14)] centered on the various atoms of the molecule as the basis functions. (For an alternative choice, see Section 15.4.) The procedure used to find the coefficients Cj, of the basis functions in each SCF MO is discussed in Section 14.3. To have a complete set of AO basis functions, an infinite number of Slater orbitals are needed, but the true molecular Hartree-Fock wave function can be closely approximated with a reasonably small number of carefully chosen Slater orbitals. A minimal basis set for a molecular SCF calculation consists of a single basis function for each inner-sheU AO and each valence-shell AO of each atom. An extended basis set is a set that is larger than a minimal set. Minimal-basis-set SCF calculations are easier than extended-basis-set calculations, but the latter are much more accurate. 

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A minimal basis set consists of one basis function for each inner shell and each valence AO. Examples of minimal- and extended-basis-set SCF wave functions for diatomic molecules were given in Sections 13.17 and 13.18. SCF wave functions give pretty accurate molecular geometries and dipole moments, but very inaccurate dissociation energies, due to improper behavior as R oo. 

Ransil, B. J. 1960a. Studies in molecular stmcture. II. LCAO-MO-SCF wave functions for selected first-row diatomic molecules. Reviews of Modem Physics 32 245-254. 

A key development in quantum chemistry has been the computation of accurate self-consistent-field wave functions for many diatomic and polyatomic molecules. The principles of molecular SCF calculations are essentially the same as for atomic SCF calculations (Section 11.1). We shall restrict ourselves to closed-shell configurations. For open shells, the formulas are more complicated. 

To overcome the deficiencies of the Hartree-Fock wave function (for example, improper behavior R oo and incorrect values), one can introduce configuration interaction (Cl), thus going beyond the Hartree-Fock approximation. Recall (Section 11.3) that in a molecular Cl calculation one begins with a set of basis functions Xi, does an SCF calculation to find SCf occupied and virtual (unoccupied) MOs, uses these MOs to form configuration (state) functions writes the molecular wave function i/ as a linear combination 2/ of the configuration functions, and uses the variation method to find the ft, s. In calculations on diatomic molecules, the basis functions can be Slater-type AOs, some centered on one atom, the remainder on the second atom. 

The Cl procedure just described uses a fixed set of orbitals in the functions An alternative approach is to vary the forms of the MOs in each determinantal function O, in (1.300), in addition to varying the coefficients c,. One uses an iterative process (which resembles the Hartree-Fock procedure) to find the optimum orbitals in the Cl determinants. This form of Cl is called the multiconfiguration SCF (MCSCF) method. Because the orbitals are optimized, the MCSCF method requires far fewer configurations than ordinary Cl to get an accurate wave function. A particular form of the MCSCF approach developed for calculations on diatomic molecules is the optimized valence configuration (OVC) method. 

In discussing H2 and H2, we saw how hybridization (the mixing of different AOs of the same atom) improves molecular wave functions. There is a tendency to think of hybridization as occurring only for certain molecular geometries. The SCF calculations make clear that all MOs are hybridized to some extent. Thus any diatomic-molecule linear combination of li, 2s, 2po, 3s, 3po, 3do,... AOs of the separated atoms. 

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