Hartree-Fock Orbitals and

To see what the solution of the above discussed nonlinear equations actually involves, let us examine these expressions in more detail for a case in which the spin-orbitals. . .., are eigenfunctions of a 

The commutator expansion of exp(-T)// exp(T) in Eq. (4.12) given in Eq. (4.13) demonstrates in an elegant manner that when Eq. (4.13) is projected against low-order excitations CV.V. = 0° y par s n, it gives equations that are at most quartic in the cluster amplitudes However, it 

Expanding the exponential then allows one to see that the only nonvanishing contributions are contained in 

In the next sections we describe how solutions may be obtained to Eq. (4.19) and we discuss the relationship of the solution thereby obtained to results ofMBPT. 

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For the N-electron species whose Hartree-Fock orbitals and orbital energies have been determined, the total SCF electronic energy can be written, by using the Slater-Condon rules, as ... 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

When forming the one-determinantal wavefunction with Kohn-Sham orbitals instead of Hartree-Fock orbitals, and taking the expectation value of the Hamiltonian, one obtains... 

Mitroy et al. (1984) carried out an extensive configuration-interaction calculation of the structure amplitude (q/ 0) for correlated target and ion states. The long-dashed curve in fig. 11.7(a) shows their momentum distribution multiplied by 2. They found that the dominant contribution came from the pseudo-orbital 3d, calculated by the natural-orbital transformation. Pseudo-orbitals are localised to the same part of space as the occupied 3s and 3p Hartree—Fock orbitals and therefore contribute to the cross section at much higher momenta than the diffuse Hartree—Fock 3d and 4d orbitals. The measurements show that the 4d orbital has a larger weight than is calculated by Mitroy et al, who overestimate the 3d component. 

The momentum distributions for the 3s ground state and the 3pi state are shown in fig. 11.15. They are compared with the momentum distributions calculated using Hartree—Fock orbitals and folding in the experimental momentum resolution function. Because the 3s and 3p orbitals are very diffuse in coordinate space the momentum profile is well within the p=l limit of validity of the plane-wave impulse approximation. 

Figure 3. Many to one correspondence between wavefunctions in Sn and one-particle densities is the Hartree-Fock orbit and is the Hohenberg-Kohn...

In evaluating correlation corrections to one-particle densities and one-electron expectation values, one must realize that single excitations only vanish to first order, but not to higher orders. This implies that the first natural orbitals differ somewhat from the Hartree-Fock orbitals, and since the singly excited configurations enter linearly in their expansion coefficient, they can have more influence on the density than doubly excited ones, even if their coefficients are smaller. 

In standard coupled-cluster theory, we use the Hartree-Fock orbitals and then determine a set of nonzero single-excitation amplitudes together with higher-excitation amplitudes. Alternatively, we may use cxp Ti) to generate an orbital transformation to a basis in which the single-excitation amplitudes vanish. Since (13.8.1) and (13.8.2) generate the same state to first order, we may use exp(— ) rather than exp(ri) to generate this orbital transformation, as done in orbital-optimized coupled-cluster (OCC) theory [29,301. The OCC ansatz for the wave function is... 

STOs, four 2s STOs and four 2p STOs [18] see Table 6.5. Indeed, in a radial-distribution plot, the numerical Hartree-Fock orbitals and the oibitals in Table 6.S would be indistinguishable. 

Let us see how we may generate a segmented basis set from the primitive functions in Table 8.2. We would like to combine these functions into a smaller set of contracted GTOs that reproduces as accurately as possible the Hartree-Fock orbitals and which at the same time has the flexibility to describe the changes that occur in the valence region upon formation of chemical bonds. Clearly, some compromises must be made since an accurate representation of the Hartree-Fock orbitals requires the use of generally contracted GTOs. 

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