10/10 MCSCF

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

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]. 

In the MPPT/MBPT method, once the reference CSF is chosen and the SCF orbitals belonging to this CSF are detennined, the wavefiinction T and energy E are detennined in an order-by-order maimer. The perturbation equations determine what CSFs to include and their particular order. This is one of the primary strengdis of this technique it does not require one to make fiirtlier choices, in contrast to the MCSCF and Cl treatments where one needs to choose which CSFs to include. 

These approaches provide alternatives to the conventional tools of quantum chemistry. The Cl, MCSCF, MPPT/MBPT, and CC methods move beyond the single-configuration picture by adding to the wavefimction more configurations whose amplitudes they each detennine in their own way. This can lead to a very large number of CSFs in the correlated wavefimction and, as a result, a need for extraordinary computer resources. 

The value of detennines how much computer time and memory is needed to solve the -dimensional Sj HjjCj= E Cj secular problem in the Cl and MCSCF metiiods. Solution of tliese matrix eigenvalue equations requires computer time that scales as (if few eigenvalues are computed) to A, (if most eigenvalues are... 

Methods that are based on making the fiinctional (T // T ) / ( T T ) stationary yield upper bounds to the lowest energy state having the synnnetry of the CSFs in T. The Cl and MCSCF methods are of this type. 

One can, for example, express T in temis of a superposition of configrirations = Y.jCj whose amplitudes Cj have been detemiined from an MCSCF, Cl or MPn calculation and express Q in temis of second-quantization operators Offt that cause single-, double-, etc, level excitations (for the IP (EA)... 

CAS-MCSCF Yes/Yes A/ transformed integrals to solve for Cl energy many iterations also i... 

Dalgaard E and J0rgensen P 1978 Optimization of orbitals for multieonfigurational referenee states J. Chem. Phys. 69 3833-44 Jensen H J Aa, J0rgensen P and A gren H 1987 Effieient optimization of large seale MCSCF wave funetions with a restrieted step algorithm J. Chem. Phys. 87 451 -66... 

Lengsfield B H III and Liu B 1981 A seeond order MCSCF method for large Cl expansions J. Chem. Phys. 75 478-80... 

Chaban G, Schmidt M W and Gordon M S 1997 Approximate second order methods for orbital optimization of SCF and MCSCF wavefunctlons Theor. Chim. Acta 97 88... 

Werner H-J and Meyer W 1981 A quadratically convergent MCSCF method for the simultaneous optimization of several states J. Chem. Phys 74 5794... 

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). 

CASSCF is a version of MCSCF theory in which all possible configurations involving the active orbitals are included. This leads to a number of simplifications, and good convergence properties in the optimization steps. It does, however, lead to an explosion in the number of configurations being included, and calculations are usually limited to 14 elections in 14 active orbitals. 

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. 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

The Seetion on More Quantitive Aspects of Electronic Structure Calculations introduees many of the eomputational ehemistry methods that are used to quantitatively evaluate moleeular orbital and eonfiguration mixing amplitudes. The Hartree-Foek self-eonsistent field (SCF), eonfiguration interaetion (Cl), multieonfigurational SCF (MCSCF), many-body and Moller-Plesset perturbation theories. 

The multiconfigurational self-consistent field ( MCSCF) method in whiehthe expeetation value < T H T>/< T T>is treated variationally and simultaneously made stationary with respeet to variations in the Ci and Cy,i eoeffieients subjeet to the eonstraints that the spin-orbitals and the full N-eleetron waveflmetion remain normalized ... 

The configuration interaction (CI) method in whieh the LCAO-MO eoeffieients are determined first (and independently) via either a single-eonfiguration SCF ealeulation or an MCSCF ealeulation using a small number of CSFs. The CI eoeffieients are subsequently determined by making the expeetation value < F H F >/< F I F >... 

A. Variational Methods Such as MCSCF, SCF, and Cl Produce Energies that are Upper Bounds, but These Energies are not Size-Extensive... 

This characteristic is commonly referred to as the bracketing theorem (E. A. Hylleraas and B. Undheim, Z. Phys. 759 (1930) J. K. E. MacDonald, Phys. Rev. 43, 830 (1933)). These are strong attributes of the variational methods, as is the long and rich history of developments of analytical and computational tools for efficiently implementing such methods (see the discussions of the CI and MCSCF methods in MTC and ACP). 

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. 

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