Density perturbation

Kierlik E and Rosinberg M L 1993 Perturbation density functional theory for polyatomic fluids III application to hard chain molecules in slitlike pores J Chem. Phys. 100 1716... 

Sychrovsky, V., Grafenstein, J., Cremer, D., 2000, Nuclear Magnetic Resonance Spin-Spin Coupling Constants from Coupled Perturbed Density Functional Theory , J. Chem. Phys., 113, 3530. 

We consider a model for the pump-probe stimulated emission measurement in which a pumping laser pulse excites molecules in a ground vibronic manifold g to an excited vibronic manifold 11 and a probing pulse applied to the system after the excitation. The probing laser induces stimulated emission in which transitions from the manifold 11 to the ground-state manifold m take place. We assume that there is no overlap between the two optical processes and that they are separated by a time interval x. On the basis of the perturbative density operator method, we can derive an expression for the time-resolved profiles, which are associated with the imaginary part of the transient linear susceptibility, that is,... 

By applying standard iterative procedures, all the perturbed density matrices can be analytically computed and thus also the electronic component of the effective properties (2.171)-(2.173), namely we have ... 

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

We write the perturbed density matrices to second order as... 

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 two-electron integrals in equations (16), (19) and (20) can be rewritten in terms of the densities p(AA r), p(SS r), and transition densitiesp(AA r) and p(SS r) that allow us to extract the physical meaning of these terms. The first one describes the electrostatic interaction between the unperturbed charge densities of A, p(AAlr) and the perturbed density of S, p(SS r), that is the contribution to the polarization energy of B in the field of A, and viceversa for A in the field of S. The second, (equation (20)), the dispersion term, contains contributions of the perturbed densities of the two systems, so that the numerators of equation (19) and equation (20) become ... 

To put the definition of this property into direct correspondence with the definition of other atomic properties, as one for which the property density at r is determined by the effect of the field over the entire molecule, we express the perturbed density in terms of the first-order corrections to the state function. This is done in a succinct manner by using the concept of a transition density (Longuet-Higgins 1956). The operator whose expectation value yields the total electronic charge density at the position r may be expressed in terms of the Dirac delta function as... 

The first-order correction to the charge density is independent of the choice of origin for the dipole moment operator, as the extra contribution to Do arising from a shift 5R in origin is — e5R<0 /c>, a term which vanishes because of the orthogonality of the zero-order states. The expression for the electric polarizability density (r) using the above expression for the perturbed density is... 

Sychrovsky et performed, for the first time, a complete implementation of coupled perturbed density functional theory (CPDFT) for the calculation of spin-spin coupling constants with pure and hybrid DFT. They analyzed the dependence of DFT with respect to the calculation of coupling constants on the exchange-correlation (XC) functionals used. They demonstrated the importance of electron correlation effects and showed that the hybrid functional leads to the best accuracy of calculated spin-spin... 

For monochromatic perturbations, the integrations are replaced by summations. Using the expansions Eqs. (45) and (46), we introduce the perturbed density matrices up to third order as... 

The perturbed densities are given by Eqs. (54) except that the contribution from to is ignored in accordance with the 2n +1 rule, as indicated by the bar in and Next, we expand according to Eq. (160) and obtain... 

Rg. M. Contour maps of the second-order perturbation densities in the H(ls)-H(ls) long-range interaction system. Nucleus b is located at a large distance R to the right, (a). ... 

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... 

From the equations above, it should be relatively obvious that calculation of the perturbed density matrices requires essentially the same steps as the calculation of the unperturbed density matrix. The only difference is that the unperturbed amplitudes and integrals in the latter are replaced sequentially (and individually) by their perturbed counterparts. As a result two, rather than one, matrix multiplications are required for their evaluation. 

Explicit dependency on the perturbation factor is thus necessary on both perturbed density and energy, in order the involved derivatives dX/dpx and d Ei)/dX be appropriately formulated and combined in working Eqs. 1.34 and 1.35. 

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