Hartree-Fock functions

General expressions for the force constants and dipole derivatives of molecules are derived, and the problems arising from their practical application are reviewed. Great emphasis is placed on the use of the Hartree-Fock function as an approximate wavefunction, and a number... 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

For practical purposes two different approaches have been used. If the nuclear framework has a center with high degree of symmetry, it may be convenient to expand the Hartree-Fock functions fk(r) in terms of spherical harmonics Ylm(6, q>) around this center ... 

Instead of using numerical integration, one can expand the unknown Hartree-Fock functions y)k(r) in terms of a fixed complete basic set... 

We note that the virial theorem is automatically fulfilled in the Hartree-Fock approximation. This result follows from the fact that the single Slater determinant (Eq. 11.38) built up from the Hartree-Fock functions pk x) satisfying Eq. 11.46 is the optimum wave function of this particular form, and, since this wave function cannot be further improved by scaling, the virial theorem must be fulfilled from the very beginning. If we consider a stationary state with the nuclei in their equilibrium positions, we have particularly Thf = — Fhf, and for the correlation terms follows consequently that... 

In considering the Hartree-Fock energy Z HF given by Eq. 11.45, we observe that, even for atoms, there are actually only a few such energies published in the literature and that, even in simple cases, there may be discrepancies between the results of different authors. The point is that the atomic Hartree-Fock functions are usually tabulated with only three decimal figures, and this numerical accuracy is often not sufficient for obtaining the accuracy desired... 

If tp(r) is the basic Hartree-Fock function or molecular orbital for H2, the total wave function may be approximated by a single determinant ... 

For systems containing three or more electrons very little is so far known about the foundation for the method of correlated wave functions, and research on this problem would be highly desirable. It seems as if one could expect good energy results by means of a wave function being a product of a properly scaled Hartree-Fock function and a correlation factor" containing the interelectronic distances ru (Krisement 1957), but too little is known about the limits of accuracy of such an approach. 

This theorem follows from the antisymmetry requirement (Eq. II.2) and is thus an expression for Pauli s exclusion principle. In the naive formulation of this principle, each spin orbital could be either empty or fully occupied by one electron which then would exclude any other electron from entering the same orbital. This simple model has been mathematically formulated in the Hartree-Fock scheme based on Eq. 11.38, where the form of the first-order density matrix p(x v xx) indicates that each one of the Hartree-Fock functions rplt y)2,. . ., pN is fully occupied by one electron. 

From the practical point of view, it may be more feasible to avoid the introduction of the virtual Hartree-Fock functions of discrete and continuous type and to use only the ordinary SCF functions xplt xp2, xp3,. . ., xpN. In order to obtain a complete basis, we will then add a conveniently chosen discrete subset q>N+1, fpN+3,. .. of functions orthogonal to the SCF functions so... 

The difficulty of determining the Half-Projected Hartree-Fock function has somewhat hampered its utilization [3-10]. Some calculations, however, exist in literature. At present time, because of the increasing computing facilities, as well as the introduction of more powerful convergence techniques, the HPHF model is expected to play a more important role, especially in the field of medium size molecules, in which the use of more sophisticated procedure are not yet possible [9-10]. 

Since rigorous theoretical treatments of molecular structure have become more and more common in recent years, there exists a definite need for simple connections between such treatments and traditional chemical concepts. One approach to this problem which has proved useful is the method of localized orbitals. It yields a clear picture of a molecule in terms of bonds and lone pairs and is particularly well suited for comparing the electronic structures of different molecules. So far, it has been applied mainly within the closed-shell Hartree-Fock approximation, but it is our feeling that, in the future, localized representations will find more and more widespread use, including applications to wavefunctions other than the closed-shell Hartree-Fock functions. 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

Equation (211) is obviously satisfied for all if <1) is a closed-shell Hartree-Fock function. We get from Eq. (212). The matrix elements (4> [y,Xi] ) vanish (for <1) a Slater determinant), except for Xk = a , a two-particle operator. 

We see the the Hartree-Fock wave functions, 4>o and i, do not, in general, satisfy orthogonality constraints analogous to those obeyed by the exact wave functions. However, we may impose constraints upon the Hartree-Fock function without loss of generality so that, for example. 

The imposition of the orthogonality constraint (14) to an approximate lower state wave function, such as the Hartree-Fock function, does not, in general yield an excited state energy which is an upper bound to the exact excited state energy. An upper bound to the excited state energy is obtained if we impose the additional constraint... 

We now consider the theoretical calculation of excited-state wave functions. This is more difficult than ground-state calculations because we are dealing with open-shell configurations. The Hartree-Fock equations for a state of an open-shell configuration have a more complicated form than for closed shells, and there exist close to a dozen different approaches to excited-state Hartree-Fock calculations. As noted earlier, the Hartree-Fock wave function for a closed-shell state is a single determinant, but for open-shell states, we may have to take a linear combination of a few Slater determinants to get a Hartree-Fock function that is an eigenfunction of S and Sz and has the correct spatial symmetry. 

The simplest approximate wavefunction for an open-shell molecule is the spin-unrestricted Hartree-Fock function... 

The remaining error in the dipole moment Green8 attributes to lack of highly excited configurations. For open-shell molecules it is probable that Hartree-Fock results will be unreliable (see above) and a limited amount of Cl will be essential. Thus even for a Hartree-Fock function the calculated one-electron properties may not agree well with experiment (it should be remembered that, in the most favourable cases where the substance can be studied in a molecular-beam spectrometer and the dipole moment obtained from Stark effect measurements, the experimental error is much less than 0.001 D).28... 

The method is based on the following procedure.33 All possible doubly excited configurations are generated from the Hartree-Fock function and their contributions to the second-order Rayleigh-Schrodinger perturbation theory energy computed. Approximately 100 of the most important are used for a Cl calculation, all singly... 

In a few cases, the wave-function F of a monatomic entity can be used for calculating a, e.g. 4.5 bohr3 for the hydrogen atom, or 0.205 A3 for the helium atom in agreement with the experimental value. Gaseous H does not have a Hartree-Fock function stable relative to spontaneous loss of an electron, and it is necessary to introduce correlation effects in order to calculate a which is said to be 31 A3. The value 1.8 A3 for H(-I) in Table 2 derives from NaCl-type LiH, NaH and KH. The anion B2Hg2 has a = 6.3 A3 to be compared with the isoelectronic C2H6 4.47 A3. Since CH4 has a =... 

The Hylleraas function, with its improved properties as compared to a Hartree-Fock function, is called a correlated wavefunction, because it takes into account the mutual electron-electron interaction much better, and the motion of electrons beyond a mean-field average is termed correlated motion or the effect of electron correlations. (The definition of electron correlation is used here in the strict terminology. The mean-field average of electron-electron interactions is frequently also called electron correlation.) Comparing equ. (1.20) with equ. (1.16b) one has... 

FIGURE 3.4 Transformation of the valence orbitals of BeH2, from canonical MOs (left-hand side) to localized bond orbitals (right-hand side). This transformation leaves the polyelectronic Hartree-Fock function unchanged. 

In addition to the energies, the values predicted by these SCF calculations for a number of the one-electron properties of the water molecule are also compared by Kern and Karplus.6 As anticipated, the values predicted for these properties by the near Hartree-Fock functions are in good agreement with the experimental values. The molecular dipole moment, for example is calculated to be 2.054 D by the best STO set compared with an experimental value of 1.884 D.57 (Hartree-Fock estimates of molecular dipole moments are generally too large by ca. 0.2 D.)... 

---