Hartree Fock momentum

Hartree-Fock (momentum space) 139 natural orbitals 27... 

Fig. 11.4 illustrates the momentum profiles of the other ion states observed in a later experiment with better energy resolution than that of fig. 11.2. All these states have momentum profiles of essentially the same shape. They are thus identified as states of the same orbital manifold, for which the experiment obeys the criterion for the validity of the weak-coupling binary-encounter approximation. Details of electron momentum spectroscopy depend on the approximation adopted for the probe amplitude of (11.1). The 3s Hartree—Fock momentum profiles in the plane-wave impulse approximation identify the 3s manifold. However, the approximation underestimates the high-momentum profile. 

The 3s momentum distribution confirms the validity of the approximations. The experiment confirms the 3pi Hartree—Fock momentum distribution and eliminates the distribution for the sum over magnetic substates that would apply to target atoms that are not oriented. With perfect... 

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

One Important aspect of the supercomputer revolution that must be emphasized Is the hope that not only will It allow bigger calculations by existing methods, but also that It will actually stimulate the development of new approaches. A recent example of work along these lines Involves the solution of the Hartree-Fock equations by numerical Integration In momentum space rather than by expansion In a basis set In coordinate space (2.). Such calculations require too many fioatlng point operations and too much memory to be performed In a reasonable way on minicomputers, but once they are begun on supercomputers they open up several new lines of thinking. 

With the above results, it is possible to write the expanded momentum space form of the Hartree-Fock equations ... 

In addition, since the HPHF wavefunction exhibits a two-determinantal form, this model can be used to describe singlet excited states or triplet excited states in which the projection of the spin momentum Ms=0. The HPHF approximation appears thus as a simple method for the direct determination of excited states (with Afs=0)such as the usual Unrestricted Hartree Fock model does for determining triplet excited states with Ms = 1. 

Secondly, correlations in the initial state can lead to experimental orbital momentum densities significantly different from the calculated Hartree-Fock ones. Figure 3 shows such a case for the outermost orbital of water, showing how electron-electron correlations enhance the density at low momentum. Since low momentum components correspond in the main to large r components in coordinate space, the importance of correlations to the chemically interesting long range part of the wave function is evident. 

Figure 3. Comparison of the measured momentum distributions of the outermost valence orbital for wafer [6-8] with spherically averaged orbital densities from Hartree-Fock limit and correlated wave functions [6].

Electron correlation is inherently a multi-electron phenomenon, and we believe that the retention of explicit two-electron information in the Wigner intracule lends itself to its description (i). It has been well established that electron correlation is related to the inter-electronic distance, but it has also been suggested (4) that the relative momentum of two electrons should be considered which led us to suggest that the Hartree-Fock (HF) Wigner intracule contains information which can yield the electron correlation energy. The calculation of this correlation energy, like HF, formally scales as N. ... 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

Additional information on orbital type and composition is available from (e,2e) or electron momentum spectroscopy (Moore et al., 1982 see Appendix B) performed on Sip4 by Fantoni et al. (1986). Electron momentum distributions measured at various binding energies have been compared with those from ah initio Hartree-Fock-Roothaan SCF calculations using a double- wave function with a single Si 3of polarization... 

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