MCSCF wavefunctions

Such a compact MCSCF wavefunction is designed to provide a good description of the set of strongly occupied spin-orbitals and of the CI amplitudes for CSFs in which only these spin-orbitals appear. It, of course, provides no information about the spin-orbitals that are not used to form the CSFs on which the MCSCF calculation is based. As a result, the MCSCF energy is invariant to a unitary transformation among these virtual orbitals. 

All of the CSFs in the SCF (in which case only a single CSF is included) or MCSCF wavefunction that was used to generate the molecular orbitals (jii. This set of CSFs are referred to as spanning the reference space of the subsequent CI calculation, and the particular combination of these CSFs used in this orbital optimization (i.e., the SCF or MCSCF wavefunction) is called the reference function. 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

In MCSCF response theory [32] the reference state is approximated by a MCSCF wavefunction... 

At this point we should mention that we encountered instability problems in the linear response calculations for some of the MCSCF wavefunctions at intemuclear distances larger than R—S a.u. We believe those instabilities to be artifacts of the calculations because their existence or position depends on the choice of basis set, active space or number of electrons allowed in the RAS3 space. This implies that even though it might not be possible to generate... 

We report MCSCF calculations of the dipole and quadmpole polarizability tensor radial functions of LiH and HF for internuclear distance reaching from almost the unified atom to the dissociation limit. Large one-electron basis sets and MCSCF wavefunctions of the CAS type with large active spaces were employed in the calculations. 

In the approach pursued here, the recovery of correlation is perceived as a two-stage process First, the determination of a zeroth-order approximation in form of a MCSCF wavefunction that is in some way related to the full valence space and determines the molecular orbitals then, the determination of refinements that recover the remaining dynamic correlation. Section 2 of this paper deals with the elimination of all configurational deadwood from full valence spaces. In Section 3, a simple approach for obtaining an accurate estimate of the dynamic correlation is discussed. 

The number No of occupied valence SCF orbitals in a molecule is typically less than the total number Nmb of orbitals in the minimal valence basis sets of all atoms. The full valence MCSCF wavefunction is the optimal expansion in terms of all configurations that can be generated from N b molecular orbitals. Closely related is the full MCSCF wavefunction of all configurations that can be generated from Ne orbitals, where Nc is the number of valence electrons, i.e. each occupied valence orbital has a correlating orbital, as first postulated by Boys (48) and also presumed in perfect pairing models (49,50), We shall call these two types of frill spaces FORS 1 and FORS 2. In both, the inner shell remains closed. 

To illustrate how the above developments are carried out and to demonstrate how the results express the desired quantities in terms of the original wavefunction, let us consider, for an MCSCF wavefunction, the response to an external electric field. In this case, the Hamiltonian is given as the conventional one- and two-electron operators H° to which the above one-electron electric dipole perturbation V is added. The MCSCF wavefunction P and energy E are assumed to have been obtained via the MCSCF procedure with H=H°+AV, where A can be thought of as a measure of the strength of the applied electric field. 

The expressions given above for E(Z=0) and (dE/dX)o can once again be used, but with the Hellmann-Feynman form for V. Once again, for the MCSCF wavefunction, the variational optimization of the energy gives... 

It should be stressed that the MCSCF wavefunction yields especially compact expressions for responses of E with respect to an external perturbation because of the variational conditions... 

Applications of continuum solvation approaches to MCSCF wavefunctions have required a more developed formulation with respect to the HF or DFT level. Even for an isolated molecule, the optimization of MCSFCF wavefunctions represents a difficult computational problem, owing to the marked nonlinearity of the MCSCF energy with respect to the orbital and configurational variational parameters. Only with the introduction of second-order optimization methods and of the variational parameters expressed in an exponential form, has the calculation of MCSCF wavefunction became routine. Thus, the requirements of the development of a second-order optimization method has been mandatory for any successful extension of the MCSCF approach to continuum solvation methods. In 1988 Mikkelsen el ol. [10] pioneered the second-order MCSCF within a multipole continuum model approach in a spherical cavity. Aguilar et al. [11] proposed the first implementation of the MCSCF method for the DPCM solvation model in 1991, and their PCM-MCSCF method has been the basis of many extensions to more robust second-order MCSCF optimization algorithms [12],... 

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],... 

In this appendix we generalise the expressions of the diabatic quantities first introduced in Sec. 2 for the ideal case of an exact two-level problem to a more realistic description. In a normal situation, the Hamiltonian has an infinite number of eigenstates, and there is no finite number of strictly diabatic states [76] that can describe a given pair of adiabatic states [77-80]. Instead, one can define a unitary transformation of the adiabatic states generating two quasidiabatic states characterised by a residual non-adiabatic coupling, as small as possible, but never zero (see, e.g., [5,24,32-35]). In practice, the electronic Hilbert space is always truncated to a finite number of configurations. In what follows, we consider the case of MCSCF wavefunctions and make use of generalised crude adiabatic states adapted to this. 

Today, the Moller-Plesset (MP) reference hamiltonian is the established choice. For the multireference case, several different choices have been made by different authors. Each of these defines a particular flavour of multireference MP2, i.e. MR-MP2. Examples include the work of Andersson and coworkers,55 57 the work of Hirao,58 the work of Davidson,59 and that of Finley and Freed,60 although this list is by no means exhaustive. Some of these methods depend on the definition of a one-electron operator, closely analogous to the closed-shell Fock operator, which can be defined for some special types of MCSCF wavefunctions. The paper by Hirao58 cited above is typical of these. 

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