Eigenvalue following method

In the course of conventional vibrational spectroscopic calculations in internal coordinates the following eigenvalue-problem has to be solved for the molecule under consideration (FG-method terminology of Wilson, Decius, and Cross (5)) ... 

The values of Km and T2d from Eq.(36) can be obtained from the transfer function of the linearized model at the equilibrium point, applying conventional methods from the linear control theory (see [1]). In order to investigate the self-oscillating behavior, one can determine the linearized system at the equilibrium point, and the corresponding complex eigenvalues with zero real part, when the parameters Km and of the PI controller are varied. For example, taking into account Eq.(34), the Jacobian matrix of the linearized system at dimensionless set point temperature xs is the following ... 

In our first exploration of the T1 and T2 conditions [5] we obtained results of the RDM method for the ground-state energy and dipole moment for a collection of small molecules and molecular ions, both closed-shell and open-shell systems. (We don t mean closed shell in a strict sense, and we only constrained the spin and spin multiplicity eigenvalues, not the elements of the RDM.) The choice of molecules and configurations largely followed Ref. [18]—a paper that, we think, reinvigorated the classical RDM approach. We showed that the addition of the T1 and T2 conditions (T2 without the off-diagonal block X)... 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator Ay = ySlXl + Vx2x2 supplied by the first kind boundary conditions may be set up in a quite similar manner as follows it is required to find the values of the parameter A (eigenvalues) associated with nontrivial solutions of the homogeneous equation subject to the homogeneous boundary conditions... 

In the GE method the minimization of the transverse modes comes first and the maximization of the reaction mode second. We now describe a method that weights minimization and maximization equally. We assume the existence of an image function with the following properties.18 If the function to be optimized fix) has the following gradient and eigenvalues at x... 

As with the solution of other many-body electronic structure problems, determination of the unperturbed eigenvalues is numerically challenging and involves compromises in the following areas (1) approximations to the hamiltonian to simplify the problem (e.g., use of semi-empirical molecular orbital methods) (2) use of incomplete basis sets (3) neglect of highly excited states (4) neglect of screening effects due to other molecules in the condensed phase. 

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