Wave function MCSCF

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. 

MCSCF methods describe a wave function by the linear combination of M configuration state functions (CSFs), with Cl coefficients, Ck,... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

It is particularly desirable to use MCSCF or MRCI if the HF wave function yield a poor qualitative description of the system. This can be determined by examining the weight of the HF reference determinant in a single-reference Cl calculation. If the HF determinant weight is less than about 0.9, then it is a poor description of the system, indicating the need for either a multiple-reference calculation or triple and quadruple excitations in a single-reference calculation. 

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. 

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. 

These expressions are only correct for wave functions that obey the Hellmann-Feynman theorem. Flowever, these expressions have been used for other methods, where they serve as a reasonable approximation. Methods that rigorously obey the Flellmann-Feynman theorem are SCF, MCSCF, and Full CF The change in energy from nonlinear effects is due to a change in the electron density, which creates an induced dipole moment and, to a lesser extent, induced higher-order multipoles. 

The ah initio methods available are RHF, UHF, ROHE, GVB, MCSCF along with MP2 and Cl corrections to those wave functions. The MNDO, AMI, and PM3 semiempirical Hamiltonians are also available. Several methods for creating localized orbitals are available. 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

RHF to UHF, or to a TCSCF, is almost pure static correlation. Increasing the number of configurations in an MCSCF will recover more and more of the dynamical correlation, until at the full Cl limit, the correlation treatment is exact. As mentioned above, MCSCF methods are mainly used for generating a qualitatively correct wave function, i.e. recovering the static part of the correlation. 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

The acronym SEC refers to the case where the reference wave function is of the MCSCF type and tire correlation energy is calculated by an MR-CISD procedure. When the reference is a single determinant (HE) the SAC nomenclature is used. In the latter case the correlation energy may be calculated for example by MP2, MP4 or CCSD, producing acronyms like MP2-SAC, MP4-SAC and CCSD-SAC. In the SEC/SAC procedure the scale factor F is assumed constant over the whole surface. If more than one dissociation channel is important, a suitable average F may be used. 

When natural orbitals are determined from a wave function which only includes a limited amount of electron correlation (i.e. not full Cl), the convergence property is not rigorously guaranteed, but since most practical methods recover 80-90% of the total electron correlation, the occupation numbers provide a good guideline for how important a given orbital is. This is the reason why natural orbitals are often used for evaluating which orbitals should be included in an MCSCF wave function (Section 4.6). 

The wave function depends on the perturbation indirectly, via parameters in the wave function (C), and possibly also the basis functions (x). The wave function parameters may be MO coefficients (HF), state coefficients (Cl, MP, CC) or both (MCSCF). 

If the wave function is variationally optimized with respect to all parameters (HF or MCSCF, but not Cl), the last term disappears since the energy is stationary with respect to a variation of the MO/state coefficients (Ho,Pi and P2 do not depend on the parameters C). 

For variationally optimized wave functions (HF or MCSCF) there is a 2n -I- 1 rule, analogous to the perturbational energy expression in Section 4.8 (eq. (4.34)) knowledge of the Hth derivative (also called the response) of the wave function is sufficient for... 

So far everything is exact. A complete manifold of excitation operators, however, means that all excited states are considered, i.e. a full Cl approach. Approximate versions of propagator methods may be generated by restricting the excitation level, i.e. tmncating h. A complete specification furthermore requires a selection of the reference, normally taken as either an HF or MCSCF wave function. 

The RPA method may be improved either by choosing an MCSCF reference wave function, leading to the MCRPA method, or by extending the operator manifold beyond... 

The energy of a wave function containing variational parameters, like an HF (one Slater determinant) or MCSCF (many Slater determinants) wave function. Parameters are typically the MO and state coefficients, but may also be for example basis function exponents. Usually only minima are desired, although in some cases saddle points may also be of interest (excited states). 

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

Based on the same two step proeedure as presented above for C2H4 (MCSCF ealeulations followed by Schmidt orthogonalization of Rydberg functions), a systematic search was conducted by progressively incorporating groups of orbitals in the active space. Two types of wave functions proved well adapted to the problem, one for in-plane excitations, the other for out-of-plane excitations from the carbene orbital. The case of the Ai states will serve as an illustration of the general approach done for all symmetries and wave functions. 

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