Multiconfigurational SCF

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. 

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. 

When the HF wave function gives a very poor description of the system, i.e. when nondynamical electron correlation is important, the multiconfigurational SCF (MCSCF) method is used. This method is based on a Cl expansion of the wave function in which both the coefficients of the Cl and those of the molecular orbitals are variationally determined. The most common approach is the Complete Active Space SCF (CASSCF) scheme, where the user selects the chemically important molecular orbitals (active space), within which a full Cl is done. 

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. 

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 Section on More Quantitive Aspects of Electronic Structure Calculations introduces many of the computational chemistry methods that are used to quantitatively evaluate molecular orbital and configuration mixing amplitudes. The Hartree-Fock self-consistent field (SCF), configuration interaction (Cl), multiconfigurational SCF (MCSCF), many-body and Mpller-Plesset perturbation theories,... 

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. 

The standard method for selecting the 4>j is to ask for the <)>i which maximize the importance of one or more terms in the sum. This gives the self-consistent-field (SCF) or multiconfiguration SCF (MC-SCF) equations. If each < >. is expanded as a linear combination of some fixed set of basis functions f - the coefficients can be found by an extension of the Roothaan SCF equations. 

The black box situation of SCF applications has not yet been reached for the multiconfigurational SCF theory. This constitutes a major problem, since MCSCF is a much better starting point for quantum chemical calculations on many interesting chemical problems (a good example is studies of transition states for chemical reactions). A development towards more automatized procedures can consequently be expectedto take place in MCSCF theory too. 

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. 

As a first application of a new analytical gradient method employing UHF reference functions, seven different methods for inclusion of correlation effects were employed to optimize the geometry and calculate the harmonic vibrational frequencies and dipole moments of the lowest open-shell states for three simple hydrides including 3Z i SiH2228. As the degree of correlation correction increased, results approached those from the best multiconfiguration SCF calculation. 

Potential energy surfaces for dissociation of SiH2 to SiH + H and Si + H2, from various states up to 8 eV excitation energy, have been calculated by multiconfiguration SCF + multireference Cl calculations252. 

Mainly high-level correlated ab initio calculations (multiconfiguration SCF, multireference Cl, and CC) and DFT. The emphasis is on highly accurate computations. .. accurate ab initio calculations can be performed for much larger molecules than with most other programs. An unusual feature is the inclusion of... 

With respect to the multiconfiguration SCF (MCSCF), the virtues and the drawbacks of this method99-101 have been described in the literature on several occasions (see e.g. Ref.5 ). In our opinion, the MCSCF still remains much more suited... 

The Valence Bond Self-Consistent Field (VBSCF) method has been devised by Balint-Kurti and van Lenthe (32), and was further modified by Verbeek (6,33) who also developed an efficient implementation in a package called TURTLE (11). Basically, the VBSCF method is a multiconfiguration SCF procedure that allows the use of nonorthogonal orbitals of any type. The wave function is given as a linear combination of VB structures, (Eq. 9.7). 

An accurate solution for the problem can be found by a complete multiconfiguration SCF treatment, in which the expansion coefficients of Eqs. (1) and (2) are determined simultaneously using the SCF techniques, with the usual trial and error procedure. This formulation can be developed along the lines of that given by Veil-lard and Clementi (1967) for closed-shell systems with inclusion of only two-electron excitations. 

Finally, we should mention some approximate calculations on H2. Jug77 has developed a semi-empirical version of the multiconfiguration SCF (MCSCF) method, using CNDO- and INDO-type approximations, and has reported the results of a double-configuration approach to Ha. It was shown that the eigenvalues of the EHF operator have physically interpretable characteristics and follow dissociation properly. Further results of this method should be very interesting. 

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