Time-dependent SCF

Another, purely quantum mechanical approximation is the so-called time-dependent self-consistent field (TDSCF) method. For general reviews see Kerman and Koonin (1976), Goeke and Reinhard (1982), and Negele (1982). For applications to molecular systems see, for example, Gerber and Ratner (1988a,b). In the TDSCF method the wavepacket is separated according to 

Inserting (4.30) into the time-dependent Schrodinger equation and multiplication with ( r yields 

Similar equations hold for r(r t) and H CF(r t). In this way the single partial differential equation in R and r is split into two ordinary differential equations, one for each degree of freedom. 

Several improvements of the TDSCF approach have been proposed in the recent literature (Kucar, Meyer, and Cederbaum 1987 Makri and Miller 1987 Meyer, Kucar, and Cederbaum 1988 Kotler, Nitzan, and Kosloff 1988 Meyer, Manthe, and Cederbaum 1990 Campos-Martinez and Coalson 1990 Waldeck, Campos-Martinez, and Coalson 1991). 

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An approximation which is convergent to the exact solution of the time-dependent Schrddinger equation can be generated with the multi configuration time dependent SCF scheme. Therein the summation in eq. (3) is truncated to a finite number N... 

We will restrict the further considerations to the case, where only one product in the expansion of the total wave function is relevant. Instead of the MCTDSCF approximation the solution is approximated by a single product function wherein these functions are determined in a self consistent way (time dependent SCF approximation, TDSCF). The situation is similar to that where there are several electronic degrees of freedom for a molecule, but where it has been demonstrated that the a batic Bom-Oppenheimer approximation works substantially well for the description of most spectroscopic and other properties of molecules. 

The central assumption behind the derivations in this chapter has been that the frequency of the time-dependent perturbation was such that the system responded to the perturbation by vibrating with that frequency. Consideration of the effect of perturbations for which this assumption is not true will lead to a much more convenient form for the time-dependent SCF perturbation expressions, which eliminates the need to solve separate equations at each frequency and, more surprisingly, for each separate spatial form of the perturbation /. 

The equations to be solved for the time-dependent SCF first-order perturbation correction" are, in the absence of a perturbation ... 

There are no new requirements for the implementation of the time-independent or time-dependent SCF equations except the generation of the electron-repulsion integrals over the molecular orbitals. We now turn to this integral transformation problem. 

FFT = fast Fourier transform GWP = Gaussian wave-packet MCTDSCF = multiconfiguration TDSCF SQ = second quantization TDGSCF = time-dependent group SCF TDSCF = time-dependent SCF TDSE = time-dependent Schrddinger equation. 

However, if the correction terms are introduced as demonstrated above, it is just a way of approaching the limit in which the trial function is expressed as ff(r, t)X(R, t) with no restriction on the form of X(R, f), i.e., it is not for instance a GWP at all times We shall denote this limit the self-consistent field limit (SCF) or rather the time-dependent SCF limit (TDSCF). The reason for this designation is that the equations for the two wavefunctions are solved self-consistently, i.e., the TDSE for one mode involves the average interaction with the other and vice versa. Thus, if we insert a product type wavefunction r r, t)X R, t) in the TDSE, equation (7), using the expansion (1), multiply fi om the left with X(R, t) and integrate over R we get... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

While the (one-particle) Brillouin condition BCi has been known for a long time, and has played a central role in Hartree-Fock theory and in MC-SCF theory, the generalizations for higher particle rank were only proposed in 1979 [38], although a time-dependent formulation by Thouless [39] from 1961 can be regarded as a precursor. 

The use of Effective Core Potential operators reduces the computational problem in three ways the primitive basis set can be reduced, the contracted basis set can be reduced and the occupied orbital space can be reduced. The reduction of the occupied orbital space is almost inconsequential in molecular calculations, since it neither affects the number of integrals nor the size of the matrices which has to be diagonalized. The reduction of the primitive basis set is of course more important, but since the integral evaluation time is in general not the bottleneck in molecular calculations, this reduction is still of limited importance. There are some cases where the size of the primitive basis set indeed is important, e.g. in direct SCF procedures. The size of the contracted basis set is very important, however. The bottleneck in normal SCF or Cl calculations is the disc storage and/or the iteration time. Both the disc storage and the iteration time depend strongly on the number of contracted functions. 

UV spectra usually involve electronic state transitions, so that simple Hartree-Fock and DFT calculations often are not sufficient PCM has been recently extended also to multi-configurational (MC-SCF) calculations [113] and to time-dependent approaches, allowing for the description of excited states and then the prediction of the so-called solvatochromic effects on these spectra. Nuclear magnetic resonance (NMR) and electron spin resonance (EPR) spectra are even more influenced by solute-solvent interactions moreover, the interpretation of experimental data is often very difficult without the support of reliable ab initio calculation, especially for EPR which is usually applied to unstable radical species. 

Enthalpy changes may be obtained with units (English) of Btu, Btu/lb, Btu/lbmol, Btu/scf, or Btu/time depending on the available data and calculation required. 

Calculated CD curve for each conformer by De Voe coupled oscillator, 7r-electron SCF-CI-DV MO, time-dependent DFT (TDDFT)... 

Olsen, J., lorgcnsen, P. Time dependent response theory with appUcations in to self consistence field (SCF and multiconfigurational self consistent field (MCSCF) wave functions, l.F.A. PRINT, Aarhus Universitet, 1994... 

R. Ahlrichs, M. Bar, M. Haser, H. Horn, C. Koknel, Electronic structure calculations on workstation computers The program system Turbomole, Chem. Phys. Lett. 162 (1989) 165 M. Haser, R. Ahlrichs, Improvements on the direct SCF method, J. Comput. Chem. 10 (1989) 104 O. Treutler, R. Ahlrichs, J. Chem. Phys. 102 (1995) 346 R. Bauernschmitt, R. Ahlrichs, Treatment of Electronic Excitations within the Adiabatic Approximation of Time Dependent Density Functional Theory, Chem. Phys. Lett. 256 (1996) 454 S. Grimme, F. Furche, R. Ahlrichs, An improved method for density functional calculations of the frequency-dependent optical rotation, Chem. Phys. Lett. 361 (2002) 321 F. Furche,... 

The SCF, or mean-field, approximation does not include the effect of energy transfer processes between the modes. The Cl approach incorporates such effects in a time-independent framework, but as was noted in the previous section this method loses much of the simplicity and insight provided by the SCF model. The most natural extension of the SCF approximation that also describes energy transfer among the coupled modes in the system, and treats this effect by a mean-field approach, is the time-dependent self-consistent-field (TDSCF), or time-dependent mean-field, approximation. 

Time Dependent-Density Functional Theory Delta-SCF... 

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