Multiconfigurational states

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. 

Let us summarise in the effective Hamiltonian language [20] the general fea-tmes of a CP /SO method, in its simple Bloch-type version. In a first step, the scalar relativistic secular equations for states under interest are solved, and extensive Cl calculations define a determinant target space dim providing accurate energies and the corresponding multiconfigurational states of interest Om) m E In a second step, a determinant inter-... 

The above outiined peculiarities in using EUE indices for the multiconfigurational states return us to the problem of constmcting excitation localization indices for arbitrary excitations. We can proceed in many ways. In the scheme [82, 84] the operator modulus of density matrix difference, A D, is used. Namely, the normal-... 

Before continuing with the ab initio procedures, in particular the MRMP theory most employed in this chapter, the use of DPT approaches for excited states will be discussed. The implementation, known with the unfortunate name of Time-Dependent DFT (TD-DFT) approach (no time-dependency is accounted for) should be expected to be able to deal with large systems were ab initio methods become too expensive. Unfortunately, these methods fail dramatically in too many situations charge transfer states, multiconfigurational states, doubly or highly-excited states, and even introduce large and systematic errors in valence states of large... 

The remainder of Section 10.8 consists of two parts. In Section 10.8.2, we discuss the implementation of Newton s method within the framework of density-based Hartree-Fock theory, in particular the transformation of trial vectors (10.8.8). Next, in Sections 10.8.3-10.8.8, we consider Newton s method in orbital-based Hartree-Fock theory. The orbital-based treatment is more general than the density-based one in that it enables us to treat more general Hartree-Fock states, thus preparing us for the multiconfigurational states of MCSCF theory in Chapter 12. 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

In this exercise, we will introduce the Complete Active Space Multiconfiguration SCF (CASSCF) method, using it to compute the excitation energy for the first excited state of acrolein (a singlet). The CIS job we ran in Exercise 9.3 predicted an excitation energy of 4.437 eV, which is rather for from the experimental value of 3.72 eV. We ll try to improve this prediction here. 

The wave functions of the ground and excited states of lanthanides have a truly multiconfigurational character.1 Therefore, computational description of both the ground state and the low-lying excited states, which are important for magnetic behaviour, is only possible by a multiconfigurational ab initio method. In this respect, the C ASSCF method proved to be a reliable tool for the description of electronic properties of lanthanide complexes. 

Perturbative methods (CASPT2 [17], NEVPT2 [18]) add the dynamical correlation in an effective way, using multiconfigurational second-order perturbation theory on the CASSCF input states. These methods have proved to be suitable for studying problems in spectroscopy, photochemistry, and so on [19, 20]. 

H. B. Gray Multiconfiguration SCF calculations by P. J. Hay indicate that the 166 -366 energy separation is over 1 eV, and there is no evidence for intervening states that could provide a facile intersystem pathway. Thus a relatively small singlet triplet intersystem crossing rate constant is not all that peculiar. 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

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