Orbital-optimized multiconfiguration

Methods based on the use of determinants, particularly those of Balint-Kurti and co-workers [37], share the advantages of simplicity and flexibility and in spite of a certain lack of mathematical elegance they also readily admit orbital optimization and the use of multiconfiguration wavefunctions. 

Dalgaard E and J0rgensen P 1978 Optimization of orbitals for multiconfigurational reference states J. Chem. Phys. 69 3833-44 Jensen H J Aa, J0rgensen P and A gren H 1987 Efficient optimization of large scale MCSCF wave functions with a restricted step algorithm J. Chem. Phys. 87 451-66... 

The Multi-Configuration Self-Consistent Field method combines the ideas of orbital optimization through a SCF technique as in the Hartree-Fock method, and a multiconfiguration expansion of the electronic wavefunction as in the configuration interaction method. In other words, the electronic wavefunction is still expressed as a linear combination of Slater determinants but now both the coeffi-... 

An example of a multireference technique is the multiconfigurational SCF (MCSCF) approach, where the wave function is obtained by simultaneously optimizing both the molecular orbitals and the configuration coefficients, thereby blending the different resonance structures together. [28] Historically, the MCSCF approach has been used extensively to provide qualitatively accurate representations of surfaces however, this method still suffers two primary drawbacks (1) the ambiguous choice of configurations and (2) the lack of dynamical correlation. 

The most straightforward way to introduce the concept of optimal molecular orbitals is to consider a trial wavefunction of the form which was introduced earlier in Chapter 9.II. The expectation value of the Hamiltonian for a wavefunction of the multiconfigurational fomn... 

The Cl procedure just described uses a fixed set of orbitals in the functions An alternative approach is to vary the forms of the MOs in each determinantal function O, in (1.300), in addition to varying the coefficients c,. One uses an iterative process (which resembles the Hartree-Fock procedure) to find the optimum orbitals in the Cl determinants. This form of Cl is called the multiconfiguration SCF (MCSCF) method. Because the orbitals are optimized, the MCSCF method requires far fewer configurations than ordinary Cl to get an accurate wave function. A particular form of the MCSCF approach developed for calculations on diatomic molecules is the optimized valence configuration (OVC) method. 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

Variational optimization of equation (11.9), where we are concerned with only one projection of tp corresponding to a particular electronic eigenstate, has been extensively studied. There are at least two well-developed techniques for such situations, namely, the multiconfiguration SCF (MCSCF) and iterative natural spin-orbital (INSO) approaches. 

Multiconfiguration Valence Bond Methods with Optimized Orbitals... 

It is straightforward also to include a core of doubly-occupied orthonormal orbitals, which may either be taken unchanged from prior calculations or optimized, simultaneously with the cip and csJ coefficients, as linear combinations of the %p. Multiconfiguration variants of the SC wavefunction may also be generated, if required, and calculations may be performed directly for excited states. 

The present article is devoted to a discussion of a multiconfiguration approach, the Optimized Basis Set - Generalized Multiconfiguration Spin-Coupled (OBS-GMCSC) method [l]-[2], that can join the flexibility of non-orthogonal orbitals with the use of simultaneously optimized Slater-type basis functions (STFs). 

The OBS-GMCSC method offers a practical approach to the calculation of multiconfiguration electronic wavefunctions that employ non-orthogonal orbitals. Use of simultaneously-optimized Slater-type basis functions enables high accuracy with limited-size basis sets, and ensures strict compliance with the virial theorem. OBS-GMCSC wavefunctions can yield compact and accurate descriptions of the electronic structures of atoms and molecules, while neatly solving symmetry-breaking problems, as illustrated by a brief review of previous results for the boron anion and the dilithium molecule, and by newly obtained results for BH3. 

There are two types of parameters that determine the RAS wave function the Cl coefficients and the molecular orbitals. When both of them are optimized the result will be a RASSCF (CASSCF) wave function, which is a extension of the SCF method to the multiconfigurational case. Below we shall briefly show how the optimization is done in practice in most modern programs (more details can, for example, be found in Ref. [25]). 

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