Hartree-Fock uncoupled

Here 0j) and 4 j) represent occupied and virtual spatial molecular orbitals with orbital energies e, and ej, respectively. Real orbitals have been assumed for simplicity. The molecular orbitals can be expressed in terms of linear combinations of atomic orbitals 4 k) = ]C c ik X/x) the matrix elements of the respective operators can be computed in the basis of these orbitals. 

In the work of Hegstrom, Rein and Sandars [25,107] and also in the later work by Mason, Tranter, MacDermott and coworkers [108-119], by Jung-wirth, Skala and Zahradnfk [120], in the early studies by Kikuchi, Wang and Kiyonaga [121,122] as well as in the work by Bakasov, Ha and Quack [105,123,124] these matrix elements have been restricted to one-center contributions, which has been motivated by the fact that both relevant op- 

The matrix elements of the spin-orbit coupling operator have been included in these works using empirically obtained or computed spin-orbit coupling constants for an effective one electron operator. The Breit-Pauli spin-orbit coupling operator (115) with all multi-center terms was employed for the first time by Kiyonaga, Morihashi and Kikuchi [125]. 

The UCHF approach can be improved, if the energy difference Eo — Eg = Ci — j + Jij between the singly excited triplet determinant and 

While the uncoupled Hartree-Fock method and the single transition approximation have the merit of computational simplicity, they suffer, however, in particular from the usually unsatisfactory description of electronically excited states with a single-determinant wavefunction. 

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This is the expression of the uncoupled Hartree Fock polarizability per unit cell [33], Since in this case, there is no coupling between the different vertical transitions, the coupling, and thus the matrices Ai and B, are responsible for the field-induced electron reorganizational effects. 

If Slater determinants obtained from the Hartree-Fock procedure are used in equations (4) and (5), we obtain the uncoupled Hartree-Fock (UCHF) scheme because the field effects upon the electron-electron interactions are not taken into account [14-15]. To go beyond this crude approximation, the wavefimctions are built as linear combina-... 

Several papers have dealt with the evaluation of wave functions including correlation in various ways. Bimstock34 has calculated the 13C shielding constants in CH4 and several other small molecules using an approximate form of uncoupled Hartree-Fock theory and the minimal basis set wave functions of Palke and Lipscomb.35 The results were similar to those obtained earlier by Ditchfield et al.33... 

Polarizability will be dealt with first because it is the easiest of the three properties to calculate and has certainly received the most attention. Many of the conclusions also apply to x and a, which are dealt with in much less detail. In each section we have tried to pick out the most important methods and consider them in detail at the expense of the less useful methods. Thus, for example, although the variational technique of Karplus and Kolker is simpler than the other uncoupled Hartree-Fock perturbation methods, it is not a very useful technique for calculating polarizabilities. It is very useful for calculations of magnetic susceptibility, however, where many other techniques are inappropriate. 

In practice, calculations of xHF are based on the uncoupled Hartree-Fock, the finite field, and the self-consistent perturbation methods. Some workers use gauge-invariant atomic orbitals (GIAOs). A full review of the gauge invariance of SCF wavefunctions has been given by Epstein. 6... 

In early years of quantum chemistry, several theoretical papers were devoted to calculations of linear and nonlinear responses of molecules to the electric field perturbations using the Uncoupled Hartree-Fock (UCHF) method. In comparison with the CI ansatz, the UCHF is less accurate in the description of electronic structure of molecules. Since this method was of some interest in computations of NLO properties we present this method in Section 5. 

The Uncoupled-Hartree-Fock method (UCHF) [31, 32, 50, 54, 85, 88, 100, 110] is also referred to as the sum-over-orbitals (SOO) method. In tliis technique, one takes the unperturbed Hamiltonian H° as a sum of one-particle Hamiltonians ... 

More recent studies include calculations of NLO properties of polyene series [4,42] and various benzene derivatives [9, 24, 39, 105, 106, 107, 111]. Most recent calculations based on the Uncoupled Hartree-Fock scheme were devoted to large organic systems like fullerenes [103], nanolubcs [116, 117], carbon cages [37], oligomers [25] and polymers [92]. 

Bursi, R., Lankhorst, M., Feil, D. Uncoupled Hartree-Fock calculations of the polarizabihty and hyperpolarizabilities of nitrophenols. J. Comp. Chem. 16, 545-562 (1995)... 

Jacquemin, D., Champagne, B., Andre, J.-M. Molecular orbital expressions for approximate uncoupled Hartree-Fock second hyperpolarizabilities. A Pariser-Parr-Pople assessment for model polyacetylene chains. Chem. Phys. 197, 107—127 (1995)... 

Liebmann. S.P.. Moskowitz, J.W. Polarizabilities and hyperpolarizabilities of small polyatomic molecules in the uncoupled Hartree-Fock approximation. J. Chem. Phys. 54, 3622-3631 (1971)... 

Tuan, D.F.-T., Epstein, S.T., Hirschfelder, J.O. Improvements of uncoupled Hartree-Fock expec-tahons values for physical properties. J. Chem. Phys. 44, 431-433 (1966)... 

The dispersion energy (inter-monomer correlation) in the MP2 appraoch is obtained at the uncoupled Hartree-Fock level as discussed above. At longe range the result is identical to that of the polarization approach using the M011er-Plesset partitioning. No coupling between correlation effects is... 

At the moment of writing very few implementations of the theory of molecular properties at the 4-component relativistic molecular level, beyond expectation values at the closed-shell Hartree-Fock level, have been reported. The first implementation of the linear response function at the RPA level in a molecular code appears to be to MO-based module reported by Visscher et al. [97]. Quiney and co-workers [98] have reported the calculation of second-order properties at the uncoupled Hartree-Fock level (see section 5.3 for terminology). Saue and Jensen [99] have reported an AO-driven implementation of the linear response function at the RPA level and this work has been extended to quadratic response functions by Norman and Jensen [100]. Linear response functions at the DFT/LDA-level have been reported by Saue and Helgaker [101]. In this section we will review the calculation of linear and quadratic response functions at the closed-shell 4-component relativistic Hartree-Fock level. We will follow the approach of Saue and Jensen [99] where the reader is referred for further details. 

We saw earlier that a very simple form of the dispersion energy is obtained from frequency-dependent polarizabilities at the so-called uncoupled Hartree-Fock level. The sum over states appearing in second order RS perturbation theory is simply a sum over (occupied and virtual) orbitals. A first improvement of this simple model is obtained by including apparent correlation [140], i.e. by using frequency-dependent polarizabilities obtained from the TDCHF method [36,141]. This method was initially proposed in the context of the multipole expansion, but could be generalized [142-146] to charge density susceptibility functions (or polarization propagators), which avoids the use... 

In the following it will be outlined, how the parity violating potentials are computed within a sum-over-states approach, namely on the uncoupled Hartree-Fock (UCHF) level, and within the configuration interaction singles approach (CIS) which is equivalent to the Tamm-Dancoff approximation (TDA), that avoids, however, the sum over intermediate states. Then a further extension is discussed, namely the random phase approximation (RPA) and an implementation along similar lines within a density functional theory (DFT) ansatz, and finally a multi-configuration linear response approach is described, which represents a systematic procedure that... 

The large computational effort associated with the solntion of the CPHF/CPKS equations can be avoided by neglecting the change of the effective one-electron (Fock) operator cansed by the orbital perturbations. This is known in the chemistry literature as the uncoupled Hartree-Fock (UCHF) approximation, and in physics as the independent particle approximation. With this approximation, the zero-frequency propagator is given, assuming real orbitals, as... 

P.-O. Astrand and G. Karlstrbm, Mol. Phys., 77, 143 (1992). Local Polarizability Calculations with Localized Orbitals in the Uncoupled Hartree-Fock Approximation. 

In the static limit, a = 0, this is called the uncoupled Hartree-Fock approximation (UCHF) (Dalgarno, 1959), which played an important role in the early days of calculations of molecular properties. 

Going one step further and retaining only the zeroth-order contribution to the hessian matrix, i.e. brings us back to the frequency-dependent version of uncoupled Hartree-Fock, Eq. (10.5), sometimes also called the zeroth-order polarization propagator approximation (ZOPPA). 

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