Density matrix derivative coupling

In the previous equation R is the density matrix derivative with respect to Qi. It can be calculated throu a coupled perturbed Hartree-Fock (CPHF) procedure. Notice that the presence of the term, that is, of the dipole moment matrix derivative in Eq. 7.14, is due... 

Similar considerations apply to the electric dipole moment the derivatives of the dipole integrals can be easily obtained whilst the derivatives of the density matrix require the use of coupled Hartree-Fock theory (e.g. Gerratt and Mills, 1968). 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

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. 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

Both the components of the gradient and of the Hessian have to be computed at E° = 0 in the framework of the coupled HF, or KS, approach described above, they can be expressed in terms of the unperturbed density matrix, and of its derivative with respect to the static field, respectively, i.e. ... 

The corresponding PCM expressions (2.193) and (2.194) show that the same physical effects are considered the static cavity field effects are explicitly represented by the matrices m°, while the static reaction field effects are implicit in the coupled perturbed HF (or KS) equations which determine the derivative of the density matrix. 

The first derivative of the density matrix with respect to the magnetic induction (dPfiv/dBi) is obtained by solving the coupled-perturbed Hartree-Fock (or Kohn-Sham) equations to which the first derivative of the effective Fock (or Kohn-Sham) operator with respect to the magnetic induction contributes. Due to the use of GIAOs, specific corrections arising from the effective operator Hcnv describing the environment effects will appear. We refer to Ref. [28] for the PCM model and to Ref. [29] for the DPM within either a HF or DFT description of the solute molecule. 

Terms related to the derivative of the density matrix do also appear, which lead to modified coupled perturbed equations, as described in detail in ref.[106]. 

On the contrary, the density derivatives are necessary to get higher order free energy derivatives in particular the second free energy derivatives require the first derivative of the density matrix P , which can be obtained by solving an appropriate coupled perturbed Hartree-Fock (CPHF) or Kohn-Sham (CPKS) equations. 

Once again, the derivative of the density matrix P can be obtained as solution of first-order coupled-perturbed Hartree-Fock equation with derivar tive Fock matrix given by eq. (1.70), exactly as for the nuclear shielding. 

Formally, the theory can be extended beyond the Golden Rule limit by considering higher order terms in the perturbation expansion of the rate in V. Starting with the Liouville equation for the density matrix, one can derive reduced equations of motion for the state populations (generalized master equations), formally exact for arbitrary coupling [88, 99, 295, 296],... 

These rules enable to generate a complete set of topologically unique diagrams for the coupled cluster approach of interest. If one is also interested in deriving the diagrams for a generalized density matrix using the above approach, the Hamiltonian in the formalism has to be replaced with the appropriate density... 

The derivatives of the MO coefficients or the density matrix can be obtained by solving the the corresponding derivatives of the Hartree-Fock equations, i.e. the coupled perturbed Hartree-Fock (CPHF) equations. These can be solved either in the MO basis or in the AO basis. After some manipulation, the CPHF equations in the MO basis can be reduced to ... 

The Hamiltonian describes a four-level system coupled by two bichromatic laser fields and the equations of motion can be derived with the density matrix method using Liouville equations... 

Factorization of the last term on the r.h.s. of Eq. (8) is harder, but it is facilitated by availabihty of quantum chan-ical software. From the coupled perturbed Hartree-Fock part of quantum chemical programs, it is possible to extract the first-order transformation matrix p9> which is needed for evaluation of derivatives of density matrix elements. 

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