Hamiltonian operator computational quantum mechanics

The analogy of the time-evolution operator in quantum mechanics on the one hand, and the transfer matrix and the Markov matrix in statistical mechanics on the other, allows the two fields to share numerous techniques. Specifically, a transfer matrix G of a statistical mechanical lattice system in d dimensions often can be interpreted as the evolution operator in discrete, imaginary time t of a quantum mechanical analog in d — 1 dimensions. That is, G exp(—tJf), where is the Hamiltonian of a system in d — 1 dimensions, the quantum mechanical analog of the statistical mechanical system. From this point of view, the computation of the partition function and of the ground-state energy are essentially the same problems finding... 

The Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in Ti gives the kinetic energy of each electron in by computing the second derivative of the electron s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each parr of electrons, and (c) the repulsion between each pair of nuclei. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

Let us take the Hamiltonian H as the operator A. Before writing it down, let us introduce atomic units. Their justification comes from something similar to laziness. The quantities one calculates in quantum mechanics are stuffed up by some constants h. = where A is the Planck constant electron charge —e its (rest) mass mo eto- These constants appear in clumsy formulas with various powers, in the numerator and denominator (see Table of Units, end of this book). One always knows, however, that the quantity one computes is energy, length, time, etc. and knows how the unit energy, the unit length, etc. are expressed by h, e, mo. 

Since we are interested in constructing the perturbation operatortbsX is to be added to the Hamiltonian, from now on, according to the postulates of quantum mechanics (Chapter 1), we will treat the coordinates X, y, z inEq. (12.8) as operators of multiplication hy just x, y, z- In addition, we would like to treat many charged particles, not just one. because we want to consider molecules. To this end, we will sum up all the above expressions, computed for each chaiged particle, separately. As a result, the Hamiltonian for the total system (nuclei and electrons) in the electric field represents the Hamiltonian of the system without field H ) and the perturbation (jH ) ... 

The constant value, E, is termed the eigenvalue and this value is, in fact, the energy of the system in quantum mechanics. T is usually termed the wavejunction. The operator H Hamiltonian) in Equation (1), like the energy in classical mechanics, is the sum of kinetic and potential parts. Equation (1) is usually so complicated that no analytical solutions are possible for any but the simplest systems. However, numerical techniques, to be briefly discussed is this section, enable Equation (1) to be converted to an algebraic matrix eigenvalue equation for the energy, and such equations can be effectively handled by powerful computers today. 

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