Density relaxed

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

A further issue arises in the Cl solvation models, because Cl wavefunction is not completely variational (the orbital variational parameter have a fixed value during the Cl coefficient optimization). In contrast with completely variational methods (HF/MFSCF), the Cl approach presents two nonequivalent ways of evaluating the value of a first-order observable, such as the electronic density of the nonlinear term of the effective Hamiltonian (Equation 1.107). The first approach (the so called unrelaxed density method) evaluates the electronic density as an expectation value using the Cl wavefunction coefficients. In contrast, the second approach, the so-called relaxed density method, evaluates the electronic density as a derivative of the free-energy functional [18], As a consequence, there should be two nonequivalent approaches to the calculation of the solvent reaction field induced by the molecular solute. The unrelaxed density approach is by far the simplest to implement and all the Cl solvation models described above have been based on this method. 

The Cl relaxed density approach [18] should give a more accurate evaluation of the reaction field, but because of its more involved computational character it has been rarely applied in Cl solvation models. The only notably exception is the Cl methods proposed by Wiberg at al. in 1991 [19] within the framework of the Onsager reaction field model. In their approach, the electric dipole moment of the solute determining the solvent reaction field is not given by an expectation value but instead it is computed as a derivative of the solute energy with respect to a uniform electric field. 

All the alternative variants of the MPn may be implemented using a relaxed density matrix or a unrelaxed density matrix, in analogy with the Cl solvation methods. In the first case the correlated electronic density is computed as a first derivatives of the free energy, while in the second case only the MPn perturbative wavefunction amplitudes are necessary. 

Other MP2 based solvent methods consist of the Onsager MP2-SCRF [19], within a relaxed density scheme analogous to the PTDE scheme, and a multipole MP2-SCRF model [28], based on a iterative unrelaxed approach. The analytical gradients and Hessian of the free energy at MP2-PTE level, has been developed within the PCM framework [29],... 

In Equation (1.161) (or equivalently in Equation (1.165) for the nonequilibrium case) we have shown that excited state free energies can be obtained by calculating the frozen-PCM energy E s and the relaxation term of the density matrix, PA (or P 1) where the calculation of the relaxed density matrices requires the solution of a nonlinear problem in which the solvent reaction field is dependent on such densities. 

By introducing the relaxed density PA and the corresponding charges into Equations (1.161) (or (1.165)) we obtain the first-order approximation to the exact free energy of the excited state by using a linear response scheme. This is exactly what we have called the corrected Linear Response approach (cLR) [33], The same scheme has been successively generalized to include higher order effects [39],... 

R. Cammi, B. Mennucci and J. Tomasi, Second-order Moller-Plesset analytical derivatives for the polarizable continuum model using the relaxed density approach, J. Phys. Chem. A, 103 (1999) 9100. 

We conclude the analysis of the solvent effects on the ICT state by considering the dipole moments and the NBO charges [41] calculated with the relaxed density matrix (see Section 7.1.1.4). 

The PCM-TDDFT excitation energies obtained from Eq. (7-10) reflect the variations of the solute-solvent interaction in the excited states in terms of the effects of the corresponding transition densities. To overcome this limitation (see the Introduction) the PCM-TDDFT scheme may exploits the relaxed density formalism (Section 7.1.1.4) to compute, for each specific electronic state, the variation of the solute solvent-interaction in terms of the changes of the electronic density. 

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. 

The lack of solvent polarization effects on the solute electronic charge distribution should be more important at the points of the energy surface where there are loose electrons, generally around the TS point. Surprisingly, the available information is scarce. It would be easy to collect information on this point by comparing values of AGei at the TS of selected reactions obtained from the two different approximations 1 and 3 of Section eq.(6.6), i.e. with the fully relaxed density function pM,and with the rigid function P°M computed in vacuo. 

For MBPT and CC methods, evaluation of the reduced density requires determining a response vector (A) as well as T. This defines a response density p = e Oo)(o (l + A)e. In addition, we want to allow the molecular orbitals to relax. The latter consideration adds another term, p", to the one-particle density. This relaxed density, p = p -I- p", is the critical quantity in CC and MBPT analytical gradient (and property) methods. " For just the one-particle part, we have p(l) = p (l) -I- p" = D(l) which will show up again when we discuss properties. 

The solution to all the foregoing reservations lies in analytical evaluation of first-order properties. The critical quantity is the relaxed density " D(l) = [Pg.155]

Results for dipole moments are shown in Table 29. ° Clearly, coupled with good basis sets, correlated methods can do quite well for this property. Elsewhere, we have used relaxed density-based CC and MBPT methods to study spin densities and the related hyperfine coupling constants to evaluate relativistic corrections (Darwin and mass-velocity term) when impor-tant and to evaluate highly accurate electric field gradients to extract nuclear quadruple moments. 

Further development then avoided this direct evaluation and advanced a general strategy for the computation of both MBPT and CC gradients [132,133] by relying on the so-called relaxed densities, a procedure that also enables the computation of other first-order properties, such as various multipole moments. In fact, the relevant codes based on this idea were first generated and exploited by Schaefer s group [134] (for numerous applications, see Refs. [18-26] in Ref. [135]). 

The difference in jp and Dp is the incorporation of orbital relaxation for the reference determinant, so that the results for a dipole moment computed with the relaxed density matrix gives precisely the same results as if it were done by differentiating the CC energy in a finite-field calculation,... 

The generalization of the interchange theorem [103] to the correlation problem is what makes CC analytical gradient theory viable, and, indeed, routine today. Also, the introduction of the response and the relaxed density matrices provides the non-variational CC generalizations of density matrix theory that makes it almost as easy to evaluate a property as with a normal expectation value. They are actually more general, since they apply to any energy expression whether or not it derives from a wavefunction This is essential, e.g. for CCSD(T). The difference is that we require a solution for both T and A if we want to use untruncated expressions for properties, as is absolutely necessary to define proper critical points. It is certainly true that... 

Today, analytical gradients for CC methods have been accomplished for CCD [116], and by using the above relaxed density formulation, closed-shell CCSD [117] and CCSDT-1 [118]. Also for CCSD(T) [119,120]. General, open-shell, symmetry-specific CCSD [113], and CCSD(T) have been presented and are widely used in the ACES II program system [121,122]. Now analytical derivatives for the full CCSDT are available... 

The derivative of the AO relaxed density matrix is now related to the corresponding MO quantity via... 

Normally, the experimentally observed longitudinal and transverse relaxations involve zero- and single-quantum processes Am = i — y = 0 or 1). The other time-dependent density matrix elements p (t) with A i > 1 are classified as multiple-quantum coherences. Although these coherences are not directly observable, it is possible to measure every element of the relaxation density matrix by employing specially designed NMR methods. These coherences are very important for multiple-quantum experiments [13] and quantum computation NMR applications, as it will be described later. 

To compute ground state properties, the models were fully relaxed to a local minimum energy configuration at zero pressure (allowing the cell shape and volume to vary). The final relaxed densities are illustrated in Table 19.1. Three independent models were generated for each of the structures to check the consistency of the... 

This is often called the unrelaxed second-order correction to the density matrix in order to distinguish it from the relaxed density matrix, which will be... 

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