Hartree-Fock approximation curve

Let us start the discussion with Fig. 15, which presents various levels of the approach to the potential curve of F2. The Hartree-Fock approximation is especially poor in this case because it does not lead to a correct dissociation limit, i.e. to the atoms in the Hartree-Fock ground states. In the case of F2, the proper dissociation is achieved... 

One reason for interest in more accurate calculations on HF has been the measured dissociation energy Z)o(HF), which can be obtained from photoionization or photoelectron spectra. Since HF+ dissociates correctly within the Hartree-Fock approximation to H+ and F(2P) in both the X2Tl and 2S+ states, PE curves were calculated by Julienne et a/.,203 and later by Bondybey et a/.,187 both of whom obtained values of Do in good agreement with experiment for the SCF calculation. The bond length in HF+ of 1.000 bohr is less than the experimental result, which the authors call into question. Julienne et al. give a detailed discussion of the adiabatic dissociation process. HF+ was also considered in the calculations of ref. 198. Richards and co-workers have reported several calculations of the PE curves of HF+, including the A2E+ state, which is correctly predicted to be bound.204... 

Schaefer250 has gone beyond the Hartree-Fock approximation and computed the ground-state PE curve, using first-order wavefunctions.251 A contracted STO basis set has been used, with 128 configurations included. The molecule now dissociates to two oxygen atoms, and De was computed to be 4.72 eV (expt. 5.21 eV). Spectroscopic constants were usually in better agreement with experiment than a previous minimal-basis full Cl calculation. The value of Re obtained was close to the experimental value. 

As the opposite to the examples given above, we note now processes that involve a fission of electron-pair bonds. Here the change in correlation energy is extremely large and the Hartree-Fock approximation is inherently incapable of giving a reasonable account of heats of reaction, A very illustrative example is provided by potential curves of diatomic molecules. From Fig, 4,2 it is seen that for larger depar ... 

This method should lead to results which are just as accurate as the results of the methods described in the previous sections, and can be used as a check on the computed potential-energy minimum E(R ) at R = Re if fl is determined from curve-fitting of the Morse potential with the computed R and De and this leads to a wrong we and/or w, then it can be assumed that De and/or Rg are/is wrong. It is to be emphasized (12) that the Morse curve can mostly not be used with essentially ionic compounds like NaF because the attraction given by the Coulomb term extends out in space to greater distances than the Morse exponential part for these compounds many other types of potential have been postulated (e.g. the Hellmann-potential or the Bom-Landd potential (77)). The reader can try to calculate cog, etc. of NaF from the SCF— LCAO—MO calculation of Matcha (72) in the Roothaan-Hartree-Fock approximation, using the Morse curve (E = —261.38 au, R =3.628 au experimental values Rg = 3.639 au, a)g=536 cm i, >g g=3.83 cm-i). 

Figure 6. Correlation energy for ground states of two-electron atoms as a function of = 1/D. Values for real atoms Li" ", may be read off at = 1/3 (indicated by arrow). For nuclear charge Z >2, the maximum deviations from linearity are about 1%. (The curve for Z = 1 terminates at = 1/2 because the hydride ion is unbound in the Hartree-Fock approximation, and correlation energies therefore are not well-defined for D > 2.)...

Fig. 4. Potential energy curve for the ground state of the hydrogen molecule obtained by Kolos and Roo-thaan (Rev. Mod. Phys., 32, 169) (1960) using the Hartree-Fock approximation together with some energy values obtained by performing matrix Hartree-Fock calculations with a universal b set of elliptical functions.

Figure 1 This figure shows the ground-state energies of the 6-electron iso-electronic series of atoms and ions, C, iV, 0 +, etc., as a function of the atomic number, Z. The energies in Hartrees, calculated in the crudest approximation, with only one 6-electron Sturmian basis function (as in Table 1), are represented by the smooth curve, while dementi s Hartree-Fock values [10] are indicated by dots. 

Fig. 3.6 Binding energy curves for the hydrogen molecule (lower panel). HF and HL are the Hartree-Fock and Heitler-London predictions, whereas LDA and LSDA are those for local density and local spin density approximations respectively. The upper panel gives the local magnetic moment within the LSDA self-consistent calculations. (After Gunnarsson and Lundquist (1976).)...

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

The adiabatic potential energy curves for these electronic states calculated in the Born-Oppenhelmer approximation, are given in Figure 1. Since we have discussed the choice of basis functions and the choice of configurations for these multiconfiguration self-consistent field (MCSCF) computations (12) previously (] - ), we shall not explore these questions in any detail here. Suffice it to say that the basis set for Li describes the lowest 2s and 2p states of the Li atom at essentially the Hartree-Fock level of accuracy, and includes a set of crudely optimized d functions to accommodate molecular polarization effects. The basis set we employed for calculations involving Na is somewhat less well optimized than is the Li basis in particular, so molecular orbitals are not as well described for Na2 (relatively speaking) as they are for LI2. 

Fig. 11.3. The 1200 eV noncoplanar-symmetric momentum profile for the 15.76 eV state of Ar" " (McCarthy and Weigold, 1988). Plane-wave impulse approximation curves are calculated with 3p orbitals. Full curve, Hartree—Fock (Clementi and Roetti, 1984) long-dashed curve, Hartree—Fock—Slater (Herman and Skillman, 1963) short-dashed curve, minimal variational basis.

Fig. 11.4. Noncoplanar-symmetric momentum profiles at the indicated energies for the ionisation of argon to some more-strongly excited ion states above the ion ground state (Weigold and McCarthy, 1978). Full curve, plane-wave impulse approximation for the Hartree—Fock 3s orbital.

Fig. 11.5. The 1500 eV noncoplanar-symmetric momentum profiles for the argon ground-state transition (15.76 eV), first excited state (29.3 eV) and the total 3s manifold (McCarthy et ai, 1989). Hartree—Fock curves are indicated DWIA, distorted-wave impulse approximation PWIA, plane-wave impulse approximation. Experimental data are normalised to the 3p distorted-wave curve with a spectroscopic factor Si5.76(3p) = 0.95. The experimental angular resolution has been folded into the calculations.

Fig. 11.13. The 1000 eV noncoplanar-symmetric momentum profiles for lead (Frost et al., 1986). Curves show the plane-wave impulse approximation. The experiment is normalised at the peak of the 6p-manifold profile (a). The 14.6 eV and 18.4 eV states of the 6s manifold are indicated by (b) and (c). Spectroscopic factors are given in table 11.2. For (a), (b) and (c) respectively the Hartree—Fock calculation (broken curve) is normalised to multiconfiguration Dirac—Fock (solid curve) by factors 0.82, 0.70 and 0.64.

Fig. 11.15. The 800 eV noncoplanar-symmetric momentum profiles for the laser-assisted ionisation of sodium (Zheng et al, 1990). Hartree—Fock curves for the indicated states are calculated in the plane-wave impulse approximation. From McCarthy and Weigold (1991).

---