Derivatives orthonormal vectors

In this section we shall go through some of the formalism needed for the coming derivation of the optimization methods. The parameters to be varied in the energy expression (3 25) are the Cl coefficients and the molecular orbitals. We will consider these variations as rotations in an orthonormalized vector space. For example, variations of the MO s correspond to a unitary transformation of the original MO s into a new set ... 

Here the <5 s are Kronecker symbols and the primed summations extend only over nonvanishing k-vectors. In deriving this result we used the orthonormality of the plane waves Q-l/2e-ifcr... 

The form of the projection operator can be derived from the expansion of the perturbed state vector into a complete orthonormal basis set, say... 

The reciprocal cell is defined by three vectors bi, hi and bs derived from the Ui, 82 and as vectors of the direct cell, and obeying the orthonormality condition a,by = 2jt8 y. 

The sum over L corresponds to a triple summation over all values of h, k, and / in the crystal. The second derivatives of potential energy with respect to the components of q are obtained from the converged Hessian matrix at the potential energy minimum. The roots of Eq. (8) yield the frequencies ft), k) for the 37V independent 1-dimensional oscillators, or normal modes of vibration. Associated with each frequency is a polarization vector, k) which is a linear combination of the original mass-weighted atomic displacements u. The polarization vectors are orthonormal, such that = ij (here, the superscript denotes the complex conjugate). These are found by diagonalizing the matrix D(k) ... 

In the present work, we extend the above overlap treatments for the case of multiple reference vectors. This involves derivation of explicit formulae for the (bi)orthonormal sets in cases analogous to la, lb, and 2a above, with reference vectors, m > 1 in number. Construction of S and S in case... 

Bloch-equation-based multistate PT formulations, termed quasidegenerate PT (QDPT) [22], largely assume an orthonormal set of vectors in the configuration interaction (Cl) space that is partitioned for a model space and its complement. This restricts applicability to model spaces easily separable from the rest, e.g., formed by simple determinants. While determinants facilitate a transparent derivation of many-body QDPT formulae [23, 24], identifying the determinants that need to be included in the model space is not always trivial. Though complete active space (CAS) appears a simple way out, CAS-based QDPT is unfortunately prone to the so-called intruder problem, especially for large active spaces. 

The explicit evaluation of the 17 , derivatives maybe avoided by using the orthonormality constraints of the MO and by using the interchange (Z-vector) method of Handy and Schaefer properly extended to the PCM framework (Cammi et al. 1999). 

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