Eigenvector expansion

However, this is still a non-equilibrium formulation of the problem, since the chemical potentials p account for the non-equilibrium condition of nonzero bias voltage. The only additional assumption in this formulation of the tunneling problem, compared to the general formulation above, is that the leads remain in thermal equilibrium. The expression can be calculated using a standard eigenvector expansion of surface and tip Green s functions ... 

The exponential of a matrix can be defined, in the case of a matrix with a full set of eigenvectors, in terms of the eigenvector expansion. Alternatively, one may think of exp(A) as the sum of the exponential series... 

Corollary ENV2-1 (spectral decomposition of SR ) Let A be a real, symmetric matrix, A = A, and thus Hermitian. From Aw = Xjw, as both A and Xj are real, it is always possible to find a real set of mutually orthonormal eigenvectors for A. Therefore, we may write any vector v e 91 as the eigenvector expansion... 

We emphasize that the validity of equations (7) and (8) depends on that of equations (5) and (6) which reflect the fact that expansion (2) is performed on the set of the eigenvectors ofH°. 

The expansion coefficients c° are determined variationally so that 0> is one of the eigenvectors of the restriction of H to the S space with eigenvalue Eq ... 

A symmetric matrix A, can usually be factored using the common-dimension expansion of the matrix product (Section 2.1.3). This is known as the singular value decomposition (SVD) of the matrix A. Let A, and u, be a pair of associated eigenvalues and eigenvectors. Then equation (2.3.9) can be rewritten, using equation (2.1.21)... 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the F >v matrix elements depend on the Cv,i LCAO-MO coefficients which are, in turn, solutions of the so-called Roothaan matrix Hartree-Fock equations- Zv F >v Cv,i = Zv S v Cvj. One should also note that, just as F (f>j = j (f>j possesses a complete set of eigenfunctions, the matrix Fp,v, whose dimension M is equal to the number of atomic basis orbitals used in the LCAO-MO expansion, has M eigenvalues j and M eigenvectors whose elements are the Cv>i- Thus, there are occupied and virtual molecular orbitals (mos) each of which is described in the LCAO-MO form with CV)i coefficients obtained via solution of... 

This equivalence can be shown most easily by carrying out a power series expansion of the function of M (e.g., of exp(M)) and allowing each term in the series to act on an eigenvector. 

If first order with respect to [Pg.73]

Davidson introduced a different method for higher eigenvalues which also avoids the need to have the elements of H stored in any particular order. In this method the kill eigenvector of H for the ni iteration is expanded in a sequence of orthonormal vectors bi, i=l n with coefficients found as the k— eigenvector of the small matrix H with elements bTHBj. Convergence can be obtained for a reasonably small value of n if the expansion vectors b are chosen appropriately. Davidson defined... 

In this connection the transfer-matrix method (Sect. 2.7) shows its superiority to the other methods, which have been treated here. By an elaborate expansion of the eigenvectors and eigenvalues of T it was arrived at a general formula for K R (m, n), which is suitable for producing the explicit formulas of K Rj(m, n) with fixed values of n. It reads... 

The L (R) matrix is chosen such that the matrix L (R)TF L (R) = Q (R) is diagonal with elements iv. The eigenvectors of F are arranged as columns in the L (R) matrix. It should also be noticed that V B sol(S (R), R) can no longer be written as a sum of an intramolecular (gas-phase potential) and an intermolecular part as in Eq. (10.18), because the harmonic expansion of the potential around the saddle point is based on the total potential energy surface and not just on the intramolecular part. By combining Eqs (10.19), (10.21), and (10.23) we see that the absolute position coordinates of the atoms in the activated complex around the saddle point of the total potential energy surface can be written as... 

The small parameter of the expansion is the mode displacement SQi, because it is always evaluated for the mode ground state or a low-energy state. Both are very localized. Let us consider the eigenvector Qi of the mode i. Then the Hamiltonian can be expanded in a Taylor series on the displacements 5Qi. 

The problem of finding a vector is usually solved by representing the required vector as an expansion with respect to some natural set of basis vectors. Following this method one can expand the vector of the n-th order correction to the k-th unperturbed vector- 44 n terms of the solutions b p (eigenvectors) of the unperturbed problem eq. (1.51) ... 

By this, the expansion coefficients uffl are themselves of the 0-eth order in A. The restriction l / k indicates that the correction is orthogonal to the unperturbed vector. In order to get the corrections to the /c-th vector, we find the scalar product of the perturbed Schrodinger equation for it written with explicit powers of A with one of the eigenvectors of the unperturbed problem p (j k). For the first order in A we get ... 

Then as previously inserting the expansion for eigenvalues and eigenvectors in powers of A for the manifold related to the k-th degenerate eigenvalue of H(n) and equating separately the terms up to the second order in A on the left and on the right sides of eq. (1.56) we get ... 

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