Hartree-Fock theory, time-dependent

The time-dependent Hartree-Fock theory was first discussed by Dirac (1930b) and subsequently in perturbative form by Dalgamo and Victor (1966). Its relationship to time-dependent perturbation theory has been discussed by Langhoff, Epstein and Karplus (1972). 

The standard Hartree-Fock LCAO equation from Chapter 6 

The Hartree-Fock equations have to be solved by the coupled Hartree-Fock method. The following article affords a typical example. 

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 

Frequency-dependent polarizability a and second hyperpolarizability y corresponding to various third-order nonlinear optical processes have been 

A simple application of the very general approach used in earlier sections leads to the time-dependent generalization of Hartree-Fock theory. The time-dependent Hartree-Fock (TDHF) equations (Dirac, 1929) were first formulated variationally by Frenkel (1934) they are also widely used in nuclear physics (see e.g. Thouless, 1%1) under the name random-phase approximation (RPA). Since the equations describe response to a perturbation, as in Section 11.9 but now time-dependent, they will 

We start from a one-determinant wavefunction Wg of spin-orbitals assumed to be orthonormal, and consider the fluctuation of these orbitals under the influence of a perturbation operator (12.1.8) or, equivalently, a single Fourier component (12.1.14). The variation is conveniently introduced by considering i —as in Section 8.2, where A is an anti-Hermitian matrix that describes the admixture of virtual orbitals tpm into the occupied set (V i). On expanding the exponential and taking the distinct elements A (r s) as the parameters the variation yields, up to second order, 

The derivatives needed in evaluating the tensor components (12.2.12) et seq. have been given already in (11.9.7), while expressions for the tensor components themselves are available in (11.9.8)-(11.9.10). In particular, the gradient components are 

in the absence of the perturbation, H = Hq and we assume the parameters p , to be chosen so that the energy functional is stationary for W = Wg, = 0 in other words, the Brillouin condition is satisfied, from (12.5.2), and Wg must satisfy the usual SCF equations. 

We now include in the Hamiltonian the time-dependent term (12.1.14) and write down explicitly the equations for the free and forced oscillations. The parameter displacements = Pmi-Pm Pmi Pml 0 

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Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

D. P. Santry, J. Chem. Phys., 70, 1008 (1979). The Singular Points of Frequency-Dependent Polarizabilities from the Time-Dependent Hartree-Fock Theory. 

McLachlan, A.D. and Ball, M.A. (1964). Time-dependent Hartree-Fock theory for... 

Kama, S.P. (1996) Spin-unrestricted time-dependent Hartree-Fock theory, J. Chem. Phys., 104, 6590-6605. 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

A.D. McLachlan, M.A. Ball, Time-Dependent Hartree-Fock Theory for Molecules Rev. Mod. Phys. 36 (1964) 844. 

It has recently been shown [ 12] that time-dependent or linear-response theory based on local exchange and correlation potentials is inconsistent in the pure exchange limit with the time-dependent Hartree-Fock theory (TDHF) of Dirac [13] and with the random-phase approximation (RPA) [14] including exchange. The DFT-based exchange-response kernel [15] is inconsistent with the structure of the second-quantized Hamiltonian. 

K.G. Kulander, Time-dependent Hartree-Fock theory of multiphoton ionization Helium, Phys. Rev. A 36 (1987) 2726. 

H. Sekino and R. J. Bartlett, Int. ]. Quantum Chem., 43, 119 (1992). New Algorithm for High-Order Time-Dependent Hartree-Fock Theory for Nonlinear Optical Properties. 

Kama and A. T. Yeates, Eds., American Chemical Society, Washington, DC, 1996, pp. 78-101. Sum-Over-State Representation of Non-linear Response Properties in Time-Dependent Hartree-Fock Theory The Role of State Truncation. 

The basic response equation of TDDFT has the same form as that of time-dependent Hartree-Fock theory, or of the Random Phase Approximation, i.e.,... 

Eshuis et al have implemented fully propagated time-dependent Hartree-Fock theory to calculate the real time electronic dynamics of closed- and open-shell molecules in strong oscillating electric fields. This method has been illustrated on the determination of the frequency-dependent polarizability of ethylene and is shown to converge, in the weak field limit, to the same results as the linearized TDHF method. 

Time-dependent Hartree-Fock theory. On the side of the wavefunction-based methods the time-dependent Hartree-Fock (TDHF) method can be used to calculated exeitation speetra. The Hartree-Fock method writes the ground-state wavefunction of the system of interest as a single Slater determinant... 

Consistent with time-independent Hartree-Fock theory the main approximation in time-dependent Hartree-Fock theory is, that the system is represented by a single Slater determinant, which now is composed of time-dependent single-particle wavefunctions. The time-dependent Schrodinger equation that has to be solved is given in eqn (1). The time-dependent Hamiltonian consists of a static Hamiltonian and an additional time-dependent operator describing the time-dependent perturbation, e.g. an electric field, which is a sum of time-dependent single-particle potentials ... 

McLachlan, A. D., Ball, M. A. (1964). Time-dependent Hartree—Fock theory for Molecules. Rev. Mod. Phys. 36, 844-855 Cioslowski, J. (1990). Density functional reformulation of molecular orbital theories, in Quant. Chem. 21, 303-316 Tachibana, A. (1988). Shape wave in density functional theory. Int. J. Quant. Chem. 34(4), 309-323 Jones, R. O., Gunnarsson, O. (1989). The density functional formalism, its applications and prospects. Rev. Mod. Phys. 61, 689-746 Bader, R. F. W. (1990). Atoms in Molecules—A Quantum Theory, Oxford University Press, Oxford. Mortier, W. J., Ghosh, S. K., Shankar, S. (1986). Electronegativity-equalization method for the calculation of atomic charges in molecules. J. Am. Chem. Soc. 108(15), 4315 320. 

In Time-dependent Hartree—Fock theory (TDHF) (Langhoff et al., 1972) this derivation is generalized to the time-dependent field of a monochromatic linear polarized radiation in the dipole approximation Eq. (3.76). The molecular orbitals are then also time dependent and are again expanded in the unperturbed orbitals... 

Kulander KC, Devi KR, Sandhya, Koonin SE (1982) Time-dependent Hartree-Fock theory of charge exchange Application to He + + He. Phys Rev A 25 2968. doi 10.1103/PhysRevA.25.2968... 

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