MCSCF calculation

In this approach [ ], the LCAO-MO coefficients are detemiined first via a smgle-configuration SCF calculation or an MCSCF calculation using a small number of CSFs. The Cj coefficients are subsequently detemiined by making the expectation value ( P // T ) / ( FIT ) stationary. 

The orbitals from which electrons are removed can be restricted to focus attention on the correlations among certain orbitals. For example, if the excitations from the core electrons are excluded, one computes the total energy that contains no core correlation energy. The number of CSFs included in the Cl calculation can be far in excess of the number considered in typical MCSCF calculations. Cl wavefimctions including 5000 to 50 000 CSFs are routine, and fimctions with one to several billion CSFs are within the realm of practicality [53]. 

The simultaneous optimization of the LCAO-MO and Cl coefficients performed within an MCSCF calculation is a quite formidable task. The variational energy functional is a quadratic function of the Cl coefficients, and so one can express the stationary conditions for these variables in the secular form ... 

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. 

In the CI method, one usually attempts to realize a high-level treatment of electron correlation. A set of orthonormal molecular orbitals are first obtained from an SCF or MCSCF calculation (usually involving a small to moderate list of CSFs). The FCAO-MO... 

An MCSCF calculation in which all combinations of the active space orbitals are included is called a complete active space self-consistent held (CASSCF) calculation. This type of calculation is popular because it gives the maximum correlation in the valence region. The smallest MCSCF calculations are two-conhguration SCF (TCSCF) calculations. The generalized valence bond (GVB) method is a small MCSCF including a pair of orbitals for each molecular bond. 

It is possible to construct a Cl wave function starting with an MCSCF calculation rather than starting with a HF wave function. This starting wave function is called the reference state. These calculations are called multi-reference conhguration interaction (MRCI) calculations. There are more Cl determinants in this type of calculation than in a conventional Cl. This type of calculation can be very costly in terms of computing resources, but can give an optimal amount of correlation for some problems. 

A configuration interaction calculation uses molecular orbitals that have been optimized typically with a Hartree-Fock (FIF) calculation. Generalized valence bond (GVB) and multi-configuration self-consistent field (MCSCF) calculations can also be used as a starting point for a configuration interaction calculation. 

For conhguration interaction calculations of double excitations or higher, it is possible to solve the Cl super-matrix for the 2nd root, 3rd root, 4th root, and so on. This is a very reliable way to obtain a high-quality wave function for the hrst few excited states. For higher excited states, CPU times become very large since more iterations are generally needed to converge the Cl calculation. This can be done also with MCSCF calculations. 

There have also been methods designed for use with perturbation theory and MCSCF calculations. Correlation effects are necessary for certain technically difficult molecules, such as CO, N2, HCN, F2, and N2O. 

The aim of the present article is to present a qualitative deseription of the optimised orbitals of molecular systems i.e. of the orbitals resulting from SCF calculations or from MCSCF calculations involving a valence Cl we do not present here a new formal development (although some formalism is necessary), nor a new computational method, nor an actual calculation of an observable quantity. .. but merely the description of the orbitals. 

In fact, it turns out that the orbitals resulting from SCF or valence MCSCF calculations in molecules ean be described in extremely simple terms by comparing them with the RHF orbitals of the separated atoms. 

In the case of a valence MCSCF calculation the difference between the optimised orbitals and these atomic RHF orbitals simply represents the way in which the atoms are distorted by the molecular environment. Thus, this difference is closely related to the idea of atoms in molecules (1). However, here, the atoms are represented only at the RHF level, and the difference concerns only the orbitals, not the intra- atomic correlation. 

We arrive now at the main purpose of the present work to find a qualitative description of the optimum orbitals (obtained by SCF or MCSCF calculations) of molecular systems. 

In fact, a simple description of the weakly occupied orbitals resulting from valence MCSCF calculations has already been presented (12). ... 

Finally, in order to ensure an homogeneous treatment of all excited states at the variational level, the MCSCF calculation should be averaged on the states under investigation. The lowest eigenfunetions of the MCSCF Hamiltonian will provide the zeroth-order wavefunetions to build the perturbation on. 

The MCSCF calculation was performed using the configuration space described in section... 

TABLE 2. Geometrical parameters for 1,3-butadiene (C4H6), 1,3,5-hexatriene (CcIIx), 1,3,5,7-octatetraene (CgHio) and 1,3,5,7,9-decapentaene (C10H12) from n-CAS-MCSCF calculations with 6-31G basis set17... 

The dipole and quadrupole polarizability tensor components of LiH were calculated by MCSCF linear response theory with the basis set of Roos and Sadlej [57] which consists of 13s-, 8p-, 6d-, and 2f-type sets of uncontracted Gaussian functions on Li and 12s-, 8p-, and 5d-type sets of uncontracted Gaussians on H. Due to the small size of the molecule we could perform MCSCF calculations over the whole range of internuclear distances with a very large CAS 0000 520,10,10,4 p g present the tensor components, isotropic, and anisotropic values of the dipole polarizability tensor a as function... 

The results of our very large MCSCF calculations can be used to benchmark the performance of other methods in the calculation of ZPVCs. Single configuration... 

The choice of the active spaces for the MCSCF calculations on HF was based on the natural orbital occupancy numbers obtained in MP2 calculations... 

Table 4. HF dependence of the polarizability tensors (in atomic units) on the number of active orbitals employed in the °°°CAS -MCSCF calculations using the daug-cc-pVQZ basis set at three internuclear distances...

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. 

The selection of configuration state functions to be included in MCSCF calculations is not a trivial task. Two approaches which can reduce the complexity of the problem are the complete active space self-consistent-field (CASSCF) [68] and the restricted active space self-consistent-field (RASSCF) [69] approach. Both are implemented in the Dalton program package [57] and are used in this study. Throughout the paper a CASSCF calculation is denoted by i active gactive RASSCF calculation by For the active spaces of HF, H2O, and CH4... 

Table 3. Active spaces and corresponding number of determinants used in the MCSCF calculations. The notation of the spaces is inactivecASactive or iSsi RAS i. SD indicates that single and double excitations out of RAS2 into RAS3 were allowed...

A two-pronged approach has been discussed for dealing with electron correlation in large systems (i) An extension of zeroth-order full-valence type MCSCF calculations to larger systems by radical a priori truncations of SDTQ-CI expansions based on split-localized orbitals in the valence space and (ii) the recovery of the remaining dynamic correlation by means of a theoretically-based simple semi-empirical formula. 

As is the case for standard orthogonal-orbital MCSCF calculations, the optimization of VB wavefimctions can be a complicated task, and a program such as CASVB should therefore not be treated as a black box . This is true, to a greater or lesser extent, for most procedures that involve orbital optimization (and, hence, non-linear optimization problems), but these difficulties are compounded in valence bond theory by the... 

---