MCSCF time-dependent

AIMD = ab initio molecular dynamics B-LYP = Becke-Lee-Yang-Parr CCSD = coupled cluster single double excitations CVC = core-valence correlation ECP = effective core potential DF = density functional GDA = gradient corrected density approximation MCLR = multiconfigurational linear response MP2 = M0ller-Plesset second-order (MRD)CI = multi-reference double-excitation configuration interaction RPA = random phase approximation TD-MCSCF = time-dependent multiconfigurational self-consistent field TD-SCF = time-dependent self-consistent field. 

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. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

The next and necessary step is to account for the interactions between the quantum subsystem and the classical subsystem. This is achieved by the utilization of a classical expression of the interactions between charges and/or induced charges and a van der Waals term [45-61] and we are able to represent the coupling to the quantum mechanical Hamiltonian by interaction operators. These interaction operators enable us to include effectively these operators in the quantum mechanical equations for calculating the MCSCF electronic wavefunction along with the response of the MCSCF wavefunction to externally applied time-dependent electromagnetic fields when the molecule is exposed to a structured environment [14,45-56,58-60,62,67,69-74],... 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

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... 

In order to determine time-dependent molecular properties utilizing the MCSCF/MM approach it is necessary to consider the time evolution of the appropriate operators and this is done by applying the Ehrenfest s equation for the evolution of an expectation value of an operator, X... 

To date, not much use has been made of the MCSCF based EOM theories as developed in the author s group. Instead, the framework of time-dependent response theory, which can treat essentially any kind of reference wave function l0,A ) including the MCSCF variety, has superceded the EOM-based developments for such cases. It is important to keep in mind, however, that both the EOM and response function theories involve formulating and solving sets of equations whose solution (i.e. the unknown energy) is an intensive energy. 

As discussed in section 2.4, several new developments are in progress in order to make the EFP-QM interface fully viable. In addition, EFP interfaces are being built with methods that can treat excited electronic states. In addition to the existing MCSCF interface, these include Cl singles and time-dependent density functional theory in the short term and more sophisticated Cl and coupled cluster methods in the longer term. 

J. Olsen, P. Jorgensen, Time-dependent Response theory with application to SCF and MCSCF wavefunctions, in Modem Electronic Structure Theory, D.R,. Yarkony (ed.) (World Scientific, 1995). 

Barnett et al. examined the bond between Au+ and ethene using both the ECP method and (in MCSCF calculations) the spdsMCPs for gold [259]. Subsequently, a series of five platinum(II) complexes of the form (N"N"N)PtCl were studied (where N N N represents the tridentate monoanionic ligands) using the time-dependent density functional theory [260]. 

The linear response methods offer a viable alternative to the Cl procedure [38]. A time-dependent (TD) perturbation theory (e.g. involving an oscillating electric field), combined with the SCF or MCSCF method is referred to as the TD-SCF (or random phase approximation, RPA) or the TD-MCSCF (or multiconfigurational linear response, MCLR), respectively. Let us consider the time development of the dipole moment (z-component for simplicity) ... 

In this chapter we will follow now the second approach, which means that we will apply time-independent and time-dependent perturbation theory from Chapter 3 to approximate solutions of the unperturbed molecular Hamiltonian. In particular, we will illustrate this in the following for Hartree-Fock, MCSCF and coupled cluster wavefunctions. 

Contrary to response theory for exact states, in Section 3.11, or for coupled cluster wavefunctions, in Section 11.4, in MCSCF response theory the time dependence of the wavefunction is not determined directly from the time-dependent Schrodinger equation in the presence of the perturbation H t), Eq. (3.74). Instead, one applies the Ehrenfest theorem, Eq. (3.58), to the operators, which determine the time dependence of the MCSCF wavefunction, i.e. the operators hj ... 

Inserting the expression for the time-dependent MCSCF wave function, Eq. (11.36), and the perturbation expansion of the wavefunction parameters, Eq. (11.39), and separating the orders one finds for the first-order equation [see Exercise 11.5]... 

At the SCF level all methods lead to the same expressions for the response functions as obtained in the random phase approximation, in Section 10.3, with the time-dependent Hartree-Fock approximation, in Chapter 11.1, or with SCF linear response theory. The QED and time-averaged QED method for an MCSCF energy was also shown to yield the same expressions as obtained from propagator or response theory in Sections 10.4 and 11.2. 

The number of variational parameters in internally contracted MCSCF-SCEP wavefunctions rarely exceeds 10 even if large basis sets and complex reference wavefunctions are employed. In contrast to the number of variational parameters, the number of coupling coefficients depends on the number of reference configurations, and can become very large if CASSCF references are used. The main problem is, therefore, the calculation and storage of the coupling coefficients. It would be very helpful if at least part of them could be recalculated each time they are needed. As will be discussed in Section 111.I, this would also allow one to relax the contraction coefficients in each direct Cl iteration, thereby improving the quality of the wavefunction. 

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