Basic Matrix Elements

Physically measurable quantities in quantum mechanics are associated with expectation values or matrix elements of the corresponding operators. It is therefore of fundamental importance to elaborate efficient methods for the calculation of such matrix elements. One of the most attractive features of the second quantized approach is its simplicity in evaluating matrix elements. To appreciate this, one must get some practice in the formalism. 

Consider first the matrix element of the operator string a between orbitals p and lo  

It is possible to obtain this result by an even simpler consideration  

The horizontal bars i j indicate that the index p must coincide with v otherwise av cannot annihilate. Similarly, X must coincide with a. 

The evaluation is simple. The transposition v X results in an expression for which the result is already known from the previous example  

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To obtain the basic matrix element expressions, assuming orthonormal orbitals, we again start from (7.2.5), and systematically evaluate the matrix elements (7.2.7) of the spin permutations. For the moment, we restrict the discussion to the singlet structures of one orbital configuration. We write (7.2.5) as... 

Nevertheless, the examination of the applicability of the crude BO approximation can start now because we have worked out basic methods to compute the matrix elements. With the advances in the capacity of computers, the test of these methods can be done in lower and lower cost. In this work, we have obtained the formulas and shown their applications for the simple cases, but workers interested in using these matrix elements in their work would find that it is not difficult to extend our results to higher order derivatives of Coulomb interaction, or the cases of more-than-two-atom molecules. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

Since 0m is expressed as a single sum in the basic set, one would expect that the matrix elements of Hov with respect to two functions 0m and 0n would be a double sum. However, using Eq. III.95 for h — l, we find... 

Bond strengths are essentially controlled by valence ionization potentials. In the well established extended Hiickel theory (EHT) products of atomic orbital overlap integrals and valence ionization potentials are used to construct the non-diagonal matrix elements which then appear in the energy eigenvalues. The data in Table 1 fit our second basic rule perfectly. 

Such methods have, however, been developed recently. In this volume the basic theory is discussed, as well as the intricate details necessary to arrive at efficient procedures for the evaluation of the energy matrix elements between electronic wavefunctions essential for large scale Cl calculation. 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

The evaluation of matrix elements for exphcitly correlated Gaussians (46) and (49) can be done in a very elegant and relatively simple way using matrix differential calculus. A systematic description of this very powerful mathematical tool is given in the book by Magnus and Neudecker [105]. The use of matrix differential calculus allows one to obtain compact expressions for matrix elements in the matrix form, which is very suitable for numerical computations [116,118] and perhaps facilitates a new theoretical insight. The present section is written in the spirit of Refs. 116 and 118, following most of the notation conventions therein. Thus, the reader can look for information about some basic ideas presented in these references if needed. 

Infinitesimal changes of the atomic positions were analyzed above. An estimate of the magnitude of reduced matrix element (/ V /) will he obtained from calculations at small, finite distortions. Table 1 presents essential data from the calculations with GAMESS [10]. Basic displacements were introduced and bond length renormalizations were effected in order to avoid stretching contributions. This process will make it more difficult to ensure that the various forms have moved an equal amount. 

Within the old adiabatic approximation, Eq. (39) is the basic starting point. However, from here on, the various approximations diverge. For ease of discussion, we shall first still make the Condon approximation, and then give the further approximations. However, it must be kept in mind that many similar approximations are also made in papers that use a non-Condon approach. The basic premise of the Condon approximation is that the electronic part of the matrix element varies sufficiently slowly with Q so that it can be taken out of the integration over dQ. The matrix element then reduces to products of electronic and vibrational integrals. In Dirac notation... 

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