Hamiltonian operator linearity

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Hamiltonian operator, 2,4 for many-electron systems, 27 for many valence electron molecules, 8 semi-empirical parametrization of, 18-22 for Sn2 reactions, 61-62 for solution reactions, 57, 83-86 for transition states, 92 Hammond, and linear free energy relationships, 95... 

The actual form of the Hamiltonian operator hp does not have to be defined at this moment. As in standard perturbation theory, it is assumed that the solution of the electronic structure problem of the combined Hamiltonian HKS +HP can be described as the solution y/(0) of HKS, corrected by a small additional linear-response wavefunction /b//(,). Only these response orbitals will explicitly depend on time - they will follow the oscillations of the external perturbation and adopt its time dependency. Thus, the following Ansatz is made for the solution of the perturbed Hamiltonian HKS +HP ... 

The prerequisite for the creation of orientation in the aligned state can also be formulated in terms of the time reversal properties of a Hamiltonian operator which represents the perturbation. As is shown in [276, 277] the alignment-orientation conversion may only take place if the time invariant Hamiltonian is involved. For instance, the Hamiltonian operator of the linear Zeeman effect is odd under time reversal and is thus not able to effect the conversion, whilst the operator of the quadratic Stark effect is even under time reversal and, as a consequence, the quadratic Stark effect can produce alignment-orientation conversion. 

For larger molecules it is assumed that a molecular wave function, , is an anti-symmetric product of atomic wave functions, made up by linear combination of single-electron functions, called orbitals. The Hamiltonian operator, H which depends on the known molecular geometry, is readily derived and although eqn. (3.37) is too complicated, even for numerical solution, it is in principle possible to simulate the operation of H on d>. After variational minimization the calculated eigenvalues should correspond to one-electron orbital energies. However, in practice there are simply too many electrons, even in moderately-sized molecules, for this to be a viable procedure. 

The time-dependent linear response principle can be applied to the CC theory [22-24], We begin with the Hamiltonian operator H that is a sum of the usual time-independent one 7/"1 and a time-dependent perturbation g(1) ... 

The restricted Hamiltonian operator for such nuclear motions may be expressed by writing the potential as a linear combination of the completely symmetric eigenvectors [33,34] ... 

We have already studied linear two rotor molecules such as CH3 — BF2 which possesses Cau and C v rotors, but only with one degree of freedom. Let us now consider acetone-like molecules which possess a Csv frame, such as pyrocatechin, and two Cav rotors. The restricted Hamiltonian operator of such molecules is the same as that of pyrocatechin (32) except for the potential energy function which has to reflect the threefold periodicity of the rotors. 

The potential energy function of the Hamiltonian operator for acetone-like molecules may be written as a linear combination of all the Ai eigenvectors [44] ... 

In obtaining the second form, we allow the to be complex, (hough ordinarily for our purposes this would not be essential. We also make use of the linearity of the Hamiltonian operator to separate the various terms in (he expectation value of (he Hamiltonian. In particular, if we require that variations with respect to a particular uf be zero (as in Eq. 1-10), we obtain... 

Finally the state Hessian matrix M is seen from Eq. (149) to be proportional to the representation of the Hamiltonian operator in the orthogonal complement basis, but with all the eigenvalues shifted by the constant amount (0). The dimension of the matrix M will be one less than the length of the CSF expansion unless it is constructed in the linearly dependent projected basis or the overcomplete CSF expansion set basis. Since the Hamiltonian matrix must usually be constructed in the CSF basis in the MCSCF method anyway, it is most convenient if M and C are also constructed in this basis. The transformation to the projected basis, if explicitly required, involves the projection matrix (1 — cc ). The matrix M only requires the two-electron integral subset that consists of all four orbital indices corresponding to occupied orbitals. 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

See e.g., T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin, 1967 K. Jorgens and J. Weidemann, Spectral Properties of Hamiltonian Operators, Led. Notes Math. 313 (1973). 

Provided that we do not consider magnetic interactions in the Hamiltonian operator explicitly, the electronic wavefunction xlf/c can be further separated into the product of the so-called space part ei and a spin function a as shown in Equation 1.10 for a two-electron system (linear combinations of spin functions are needed for systems with more electrons). For a single electron, the spin function a = a (or ) represents the state with ms = V2 and the function /3 (or j) that with ms = V2. 

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