Wave function general MCSCF

With regard to the second point, it is important to note that an approximate wave function which is more general than those of Eqs. (2)-(5) cannot be described in terms of either bent bonds or wave function (general MCSCF, GVB-CI or Cl) will be a complicated combination of these two descriptions (as well as others, e.g., the atoms-in-molecule picture (10)) or in certain approximate wave functions the descriptions are related by a transformation and are thus in some sense equivalent (10). Hence the best one can do is decide on a criterion to measure the extent to which a particular picture is contained in the general wave function. One possible measure would be the overlap of a unique or unique bent bond description with the general wave function. 

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. 

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. 

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. 

In general, then, an MCSCF calculation involves a specification of what MOs may be occupied in the CSFs appearing in the expansion of Eq. (7.1). Given that specification, the formalism finds a variational optimum for the shape of each MO (as a linear combination of basis functions) and for the weight of each CSF in the MCSCF wave function. 

The Fock matrix (4 42) is in general not Hermitian for a non-converged MCSCF wave function. With optimized orbitals the gradient is zero. The MCSCF Fock matrix is thus Hermitian at this point on the energy surface. This condition has been used as a basis for optimization schemes in earlier developments of the MCSCF methodology. Convergence of such first order optimization schemes is, however, often poor, and they are not very much used today. 

Computationally the super-CI method is more complicated to work with than the Newton-Raphson approach. The major reason is that the matrix d is more complicated than the Hessian matrix c. Some of the matrix elements of d will contain up to fourth order density matrix elements for a general MCSCF wave function. In the CASSCF case only third order term remain, since rotations between the active orbitals can be excluded. Besides, if an unfolded procedure is used, where the Cl problem is solved to convergence in each iteration, the highest order terms cancel out. In this case up to third order density matrix elements will be present in the matrix elements of d in the general case. Thus super-CI does not represent any simplification compared to the Newton-Raphson method. 

One way of achieving size-consistency for a dissociation process is to use an MCSCF wave function as the reference. Unfortunately, as noted above, there are as yet no general multireference perturbation theory or multireference coupled-cluster treatments that can be applied to such an MCSCF reference function. For rather few electrons, as we shall see, the MRCI approach performs acceptably. 

To summarize, the first anharmonicity may be evaluated for an MCSCF wave function with second and third derivative integrals in the AO basis, first derivative integrals in the MO basis with two general and two active indices, and undifferentiated integrals with three general and one active indices. 

There has been some recent concern (8,9) however, that this bent bond description of multiple bonds derived from the GVB-PP model may be an artifact of the model. The concern takes two forms first, that the SOPP restrictions on the GVB wave function are the source of the bent bonds and the full GVB model will produce the usual <7,7r-bond description and second, if an MCSCF or Cl wave function which is more general than GVB is used, this will give back the c,7r description. 

General Atomic and Molecular Electronic Structure System a general ab initio quantum chemistry package that can compute wave functions ranging from RHF, ROHF, UHF, GVB, and MCSCF... 

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