Perturbed density matrix

To proceed further, we look at the perturbed density matrix. It was assumed to have the form... 

We have used that the second-order perturbed density matrix elements (see Eq. (70)) can be separated into components due to first- and second-order parameters, respectively,... 

Here, we have also separated the perturbed density matrix into terms that depend on first- and second-order, and third-order parameters, respectively, as... 

From (12) it can be seen that either the symmetric (X + Y) or the antisymmetric (Y X) part of the perturbed density matrix is needed. An expression for either one is readily obtained by adding and subtracting the two equations obtained from (13) to yield... 

For a perturbing electric field in the v-direction we have V = W = Dv and W — Y = 0, while for a magnetic field in the v-direction we have for the imaginary magnetic moment operator W = —V = +MV and V + W = 0. A nonzero frequency couples the symmetric and the antisymmetric part of the perturbed density matrix, whereas in the static case the two equations in (16) are not coupled. For comments on the apparent lack of symmetry for the perturbation equations for static electric and magnetic fields see [46]. 

The nonlinear interaction of light with matter is useful both as an optical method for generating new radiation fields and as a spectroscopic means for probing the quantum-mechanical structure of molecules [1-5]. Light-matter interactions can be formally classified [5,6] as either active or passive processes and for electric field based interactions with ordinary molecules (electric dipole approximation), both may be described in terms of the familiar nonlinear electrical susceptibilities. The nonlinear electrical susceptibility represents the material response to incident CW radiation and its microscopic quantum-mechanical formalism can be found directly by diagrammatic techniques based on the perturbative density matrix approach including dephasing effects in their fast-modulation limit [7]. Since time-independent (DC) fields can only induce a... 

Whereas the calculation of the first derivative is straightforward, the calculation of the second derivative involves the perturbed density matrix for the constraint term. For perturbation-independent basis and auxiliary function sets, it follows from McWeeny s self-consistent perturbation (SCP) theory for the elanents of the (closed-shell) perturbed density matrix ... 

Backsubstitution into the expression for the perturbed density matrix elements then yields... 

This important equation contains two contributions the first term is referred to as a first-order contribution since it only depends on the ground state density. The second term is a second-order contribution since it requires the knowledge of the first derivative of the density matrix with respect to an external perturbation. These two terms substitute the two first- and second-order terms in the perturbation sum of Eq. (36). It remains to be shown how the perturbed density matrix can be calculated. 

For second energy derivatives that appear in the calculation of polarizabilities, chemical hardness, van der Waals coefficients, vibrational frequencies and other second order properties, the perturbed density matrix is required. McWeeny s self-consistent perturbation (SCP) theory (Diercksen and McWeeny 1966 Dodds et al. 1977 McWeeny 1962, 2001 McWeeny and Dier-cksen 1968 McWeeny et al. 1977) represents a direct approach for the calculation of this matrix. For the clarity of the presentation we assume perturbation-independent basis and auxiliary functions and restrict ourselves to closed-shell systems. Under these conditions the elements of the perturbed density matrix are given by the SCP formalism of McWeeny et al. (1977) ... 

In the following, we focus on the determination of the density matrix as perturbed with respect to the magnetic field (P )s whereas an extension to other perturbations is straightforward. Within a linear response formalism (only terms linear in the external perturbation are considered), we can solve for the perturbed density matrix P directly... 

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