Expectation values and matrix elements

Before solving the time-independent Schrodinger equation, equation (3.9), explicitly we briefly review some of the algebraic properties of wavefunctions. All the information about a quantum-mechanical system is contained in the wavefunction f (r,t) which is usually subjected to the normalization condition 

If T is not an eigenfunction of F it can be expanded in terms of any complete set of normalized orthogonal eigenfunctions 

Thus the operator F is now represented by a square matrix with the element in the row and i column being given by jIrIi ). The quantity j r i = F is called the matrix element of the operator F. It depends not only on the form of the operator but also on the basis set w which was chosen for the representation of the operator. If the functions Wj are eigenfunctions of the operator F corresponding to the eigenvalue then we have 

We shall be concerned mainly with the wavefunctions of electrons in atoms and in this case the predominant contribution to the potential comes from the Coulomb attraction of the nucleus. This potential is. spherically symmetric and therefore V(r) is a function of tlie radial coordinate r alone, This enables the Schrodinger equation to be separated into three differential equations which involve r, 0, and (j separately. If we consider the motion of a single electron of mass m about a nucleus of mass M we can separate off the centre-of-mass motion and consider only the relative motion of the electron. In spherical polar coordinates equation C3.8) becomes (Problem 3.3) 

Since the left-hand side of equation (3.14) depends only on r and the right-hand side on 0 and ( ), both sides must equal a constant which we call X. Thus Schrodinger s equation separates into a radial equation 

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At that time the permanent electric dipolar moment po of HCl had already been estimated to be 3.59 X10 C m [128], but Dunham made no use of this value hence we leave po in symbolic form. One or other value of coefficient p2 depends on a ratio (l/p(x) 0)/(2 p(x) 0) of pure vibrational matrix elements of electric dipolar moment between the vibrational ground state and vibrationaUy excited state V = 1 or 2. We compare these data with an extended radial function derived from 33 expectation values and matrix elements in a comprehensive statistical treatment [129],... 

LeRoy, R. J. LEVEL 6.1, A Computer Program Solving the Radial Schrodinger Equation for Bound and Quasibound Levels, and Calculating Various Expectation Values and Matrix Elements, University of Waterloo Waterloo, Canada, 1996 Chemical Physics Research Report CP-555R. 

A brief summary of the mathematical notation adopted throughout this text is in order. Scalar quantities, whether constants or variables, are represented by italic characters. Vectors and matrices are represented by boldface characters (individual matrix elements are scalar, however, and thus are represented by italic characters that are indexed by subscript(s) identifying the particular element). Quantum mechanical operators are represented by italic characters if diey have scalar expectation values and boldface characters if their expectation values are vectors or matrices (or if they are typically constructed as matrices for computational purposes). The only deliberate exception to the above rules is that quantities represented by Greek characters typically are made neither italic nor boldface, irrespective of their scalar or vector/matrix nature. 

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

One apparent disadvantage of the log-derivative methods is that they do not directly give explicit wavefunctions, which are needed to calculate molecular properties (via expectation values) and spectroscopic intensities (via off-diagonal matrix elements). However, the restriction is not as serious as it might appear a finite-difference approach for extracting expectation values from coupled channel calculations has been described by Hutson, and is available as an option in the BOUND program. 

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

Since all energy-resolved observables can be inferred from appropriate expectation values of an energy-resolved wavefunction, Eq. (21) shows that the RWP method can be used to infer observables. Specific formulas for S matrix elements or reaction probabilities are given in Refs. [1] and [13]. See also Section IIIC below. 

Double-closure is the joint operation of dividing each element of the contingency table X by the product of its corresponding row- and column-sums. The result is multiplied by the grand sum in order to obtain a dimensionless quantity. In this context the term dimensionless indicates a certain synunetry in the notation. If x were to have a physical dimension, then the expressions involving x would appear as dimensionless. In our case, x represents counts and, strictly speaking, is dimensionless itself. Subsequently, the result is transformed into a matrix Z of deviations of double-closed data from their expected values ... 

From the closure relation Z j j ) (j = 1 -1 g ) < g I, the sum over the product of transition matrix elements involving p,(r) and p (r )separates into two terms, one containing the ground-state expectation value of p (r) p (r ) and the other containing the product of the expectation values of p (r) and p (r ), both in the ground state. These terms can be further separated into those containing self interactions vs. those containing interactions between distinct electrons. Then... 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

The sp-d hybridization energy is the contribution that results from turning on the hybridization matrix elements in eqn (7.25), resulting in the mixing between the NFE sp and d bands. As expected, it is negative, taking the approximately constant value of about 2 eV across the series. 

Note that we introduced the superscript CD in order to distinguish the expressions obtained by Clark and Davidson from those by Mayer, which will be given in the following marked by Ma. In a similar fashion, Mayer s partitioning of the total spin expectation value can be derived. Starting from Lowdin s expression for the total spin expectation value, Eq. (96), a one-electron basis set is introduced as in Eq. (102) and the numbers of a- and / -electrons, Na and N13, respectively, are replaced by sums over diagonal matrix elements Y (P"S)W and E (P S) w [cf. Eq. (104)], M... 

The probability is a function of the incident energy per unit time, per unit area, I (co) Aco of the incident radiation in the frequency interval between co and co + Acu. We will also refer to /(co) as the spectral intensity of the incident radiation. The matrix element represents the expectation value of the dipole moment operator between initial and final state, hcofi = Ef—Ei is Bohr s frequency condition it is related to the energies of the initial and final states, i), /), and n designates the refractive index. 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

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