Time-dependent Hartree-Fock linear response

The corrected Linear Response approach (cLR) consists in the use the TDDFT relaxed density and the corresponding apparent charges (7-38) into Eqs. (7-36) and (7-37) to obtain the first-order approximation to the state specific free energy of the excited state. The details of the implementation are described in Ref. [17], This corrected Linear Response computational scheme can be applied to the analogous of the Time Dependent Hartree-Fock approach either in the complete (Random Phase Approximation) or approximated (Tamm-Dancoff approximation or Cl singles, CIS) version. 

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. 

The supennolecule approach is used to study the linear and second-order nonlinear susceptibilities of the 2-methyl-4-nitroaniline ciystal. The packing effects on these properties, evaluated at the time-dependent Hartree-Fock level with the AMI Hamiltonian, are assessed as a function of the size and shape of the clusters. A simple multiplicative scheme is demonstrated to be often reliable for estimating the properties of two- and three-dimensional clusters from the properties of their constitutive one-dimensional arrays. The electronic absorption spectra are simulated at the ZINDO level and used to rationalize the linear and nonlinear responses of the 2-methyl-4-nitroaniline clusters. Comparisons with experiment are also provided as well as a discussion about the reliability of the global approach. 

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. 

Using the ground-state density matrix as an input, the CEO procedure - computes vertical transition energies and the relevant transition density matrices (denoted electronic normal modes ( v)mn = g c j Cn v ), which connect the optical response with the underlying electronic motions. Each electronic transition between the ground state and an electronically excited state v) is described by a mode which is represented hy K x K matrix. These modes are computed directly as eigenmodes of the linearized time-dependent Hartree—Fock equations of motion for the density matrix (eq A4) of the molecule driven by the optical field. 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

This approximation is better known as the time-dependent Hartree—Fock approximation (TDHF) (McLachlan and Ball, 1964) (see Section 11.1) or random phase approximation (RPA) (Rowe, 1968) and can also be derived as the linear response of an SCF wavefunction, as described in Section 11.2. Furthermore, the structure of the equations is the same as in time-dependent density functional theory (TD-DFT), although they differ in the expressions for the elements of the Hessian matrix E22. The polarization propagator in the RPA is then given as... 

At the SCF level all methods lead to the same expressions for the response functions as obtained in the random phase approximation, in Section 10.3, with the time-dependent Hartree-Fock approximation, in Chapter 11.1, or with SCF linear response theory. The QED and time-averaged QED method for an MCSCF energy was also shown to yield the same expressions as obtained from propagator or response theory in Sections 10.4 and 11.2. 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

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... 

The derivation by Dirac [13] of TDHF can be expressed in terms of orbital functional derivatives [12]. This derivation can be extended directly to a formally exact time-dependent orbital-functional theory (TDOFT). The Hartree-Fock operator H is replaced by the OFT operator Q = TL + vc throughout Dirac s derivation. In the linear-response limit, this generalizes the RPA equations [14] to include correlation response [17]. 

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