Time dependent coupled Hartree-Fock

The variation dJ = 0 yields the so-called time-dependent coupled Hartree-Fock equations. When the field does not break the periodic symmetry of the polymer, these equations are already formulated for a polymer. Since, however, the different cases of the effects of stationary magnetic fields on the band structure of a polymer have not yet been programmed, and therefore no calculations are available for this problem, we shall not give here the rather complicated equations obtained for the case of nonstationary magnetic fields. 

For a relativistic generalization of the time-dependent coupled Hartree-Fock equations see J. Ladik, J. Ci2ek, and P. K. Mukherjee, in Relativistic Effects in Atomsy Molecules and Solids (G. L. Malli, ed.), p. 305, Plenum Press, New York-London (1983). 

A recent quantum-chemical ab initio ACPF (averaged coupled-pair functional) calculation gave the parallel and perpendicular components of the polarizability of NH(X Z"), 0C = 11.770 ao ( = 1.744 A ) and a = 9.016 ao ( = 1.336 A ) the average molecular polarizability thus is a= (oC + 2a )/3 = 9.934 Sq ( = 1.472 A ). Slightly higher values were obtained for the two lowest excited states, a and b of NH [1]. Using the time-dependent coupled Hartree-Fock method, the frequency-dependent polarizability a(co) was calculated for the isotropic static polarizability, a value of 9.3655 a was obtained [2]. 

Visser, F. and Wormer, P. E. S. (1984). The non-empirical calculation of second order molecular properties by means of effective states. I Application to time-dependent coupled Hartree-Fock method. Mol. Phys., 52, 923-937. 

Visser, F., Wormer, P. E. S., and Stam, P. (1983). Time-dependent coupled Hartree-Fock calculations of multipole polarizabilities and dispersion interactions in van der Waals dimers consisting of He, H2, Ne, and N2. J. Chem. Phys., 79, 4973-4984. Visser, F., Wormer, P. E. S., and Stam, P. (1984). Erratum Time-dependent coupled Hartree-Fock calculations of multipole polarizabilities and dispersion interactions in van der Waals dimers consisting of He, H2, Ne, and N2 [J. Chem. Phys. 79, 4973 (1983)]. J. Chem. Phys., 81, 3755. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

Dispersion of Linear and Nonlinear Optical Properties of Benzene An Ab Initio Time-Dependent Coupled-Perturbed Hartree-Fock Study Shashi P. Kama, Gautam B. Talapatra and Paras N. Prasad Journal of Chemical Physics 95 (1991) 5873-5881... 

The simplest polarization propagator corresponds to choosing an HF reference and including only the h2 operator, known as the Random Phase Approximation (RPA). For the static case oj = 0) the resulting equations are identical to those obtained from a Time-Dependent Hartree-Fock (TDHF) analysis or Coupled Hartree-Fock approach, discussed in Section 10.5. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

The experimental measures of these molecular electric properties involve oscillating fields. Thus, the frequency-dependence effects should be considered when comparing the experimental results . Currently, there are fewer calculations of the frequency-dependent polarizabilities and hyperpolarizabilities than those of the static properties. Recent advances have enabled one to study the frequency dispersion effects of polyatomic molecules by ab initio methods In particular, the frequency-dependent polarizability a and hyperpolarizability y of short polyenes have been computed by using the time-dependent coupled perturbed Hartree-Fock method. The results obtained show that the dispersion of a increases with the increase in the optical frequency. At a given frequency, a and its relative dispersion increase with the chain length. Also, like a, the hyperpolarizability y values increase with the chain length. While the electronic static polarizability is smaller than the dynamic one, the vibrational contribution is smaller at optical frequencies. ... 

We saw in Section III that the polarization propagator is the linear response function. The linear response of a system to an external time-independent perturbation can also be obtained from the coupled Hartree-Fock (CHF) approximation provided the unperturbed state is the Hartree-Fock state of the system. Thus, RPA and CHF are the same approximation for time-independent perturbing fields, that is for properties such as spin-spin coupling constants and static polarizabilities. That we indeed obtain exactly the same set of equations in the two methods is demonstrated by Jorgensen and Simons (1981, Chapter 5.B). Frequency-dependent response properties in the... 

Quantum Chemistry (1992) which is entirely devoted to molecular non-linear optics. The foundations of time-dependent coupled-perturbed Hartree Fock theory, referred to in Section 3.1.2, are beautifully described in Sections 12.5-12.7 of Roy McWeeny s book [7]. 

S. P. Kama, Chem. Phys. Lett., 214, 186 (1993). A Direct Time-Dependent Coupled Perturbed Hartree-Fock-Rootham Approach to Calculate Molecular (Hyper)polarizabilities. 

In analogy to the derivation of the coupled Hartree-Fock equations one can then derive the time-dependent Hartree-Fock equations for the first-order expansion coefficients... 

In the first method, Frankel s variational principle is used to give equations for the changes in the Hartree-Fock orbitals caused by an applied time dependent field. These equations are called the time-dependent coupled perturbed Hartree-Fock (TDCPHF) equations. They involve matrices known as the electric and magnetic Hessians which have coulomb and exchange contributions. The first-order TDCPHF equations can be used to calculate a and but in order to calculate Y it is necessary to use second-order TDCPHF. 

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