Matrix Elements of Operators

Calculation of the energy levels of an atom or ion normally processed by first figuring the matrix elements of the electrostatic perturbation potential V, defined as  

Since the electrostatic Hamiltonian commutes with the angular momentum operators corresponding to L, S, and M the matrix elements will be diagonal in L and S (although not it t) and independent of J and M. Calculation of the matrix elements of Eq. (5.5) commences by first expanding the interaction between each pair of electrons in Legendre polynomials of the cosine of the angle Wy between the vector from the nucleus to two electrons. 

The energies and wavefunctions can be expressed in terms of certain integrals 

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Bracket (matrix element) of operator O between functions n and m Average value of O Norm of O... 

In the latter expression the matrix element of operator dq> is transformed according to the Wigner-Eckart theorem and the definition used is... 

The matrix element of operator is written in terms of the Wigner-Eckart theorem, and the integral part is denoted as... 

Recoupling coefficients are important in computing matrix elements of operators. Consider, for example, the C operators defined, for triatomic molecules, in Eq. (4.68). For three bonds (tetratomic molecules) one has... 

When evaluating matrix elements of operators for coupled systems, it is often convenient to make use of reduction formulas. These formulas reduce the evaluation of products of operators to matrix elements of individual operators. Two situations can occur. 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

The relation between the CFP with a detached electrons and the reduced matrix elements of operator q>(lNfiLS generating [see (15.4)] the <7-electron wave function is established in exactly the same way as in the derivation of (15.21). Only now in the appropriate determinants we have to apply the Laplace expansion in terms of a rows. The final expression takes the form... 

In [90] the relationship between eigenvalues of the Casimir operators of higher-rank groups and quantum numbers v, N, L, S is taken into account to work out algebraic expressions for some of the reduced matrix elements of operators (Uk Uk) and (Vkl Vkl). However, the above formulas directly relate the operators concerned, and some of these formulas are not defined by the Casimir operators of respective groups. 

The method of CFP is an elegant tool for the construction of wave functions of many-electron systems and the establishment of expressions for matrix elements of operators corresponding to physical quantities. Its major drawback is the need for numerical tables of CFP, normally computed by the recurrence method, and the presence in the matrix elements of multiple sums with respect to quantum numbers of states that are not involved directly in the physical problem under consideration. An essential breakthrough in this respect may be finding algebraic expressions for the CFP and for the matrix elements of the operators of physical quantities. For the latter, in a number of special cases, this can be done using the eigenvalues of the Casimir operators [90], however, it would be better to have sufficiently simple but universal formulas for the CFP themselves. 

Equations of this kind can also be derived for the special cases of reduced matrix elements of operators composed of irreducible tensors. Then, using the relation between CFP and the submatrix element of irreducible tensorial operators established in [105], we can obtain several algebraic expressions for two-electron CFP [92]. Unfortunately such algebraic expressions for CFP do not embrace all the required values even for the pN shell, which imposes constraints on their practical uses by preventing analytic summation of the matrix elements of operators of physical quantities. It has turned out, however, that there exist more general and effective methods to establish algebraic expressions for CFP, which do not feature the above-mentioned disadvantages. 

The method presented enables expressions to be found for submatrix elements of irreducible tensorial products of second-quantization operators for configurations of any complexity. This method provides a unified approach both to diagonal and non-diagonal (relative to the configuration) matrix elements of operators of physical quantities. 

These equations can be used to establish additional analytical relationships when dealing with the matrix elements of operators of physical quantities in the case of complex electron configurations. 

Matrix elements of operator (22.9) may be expressed in terms of the corresponding submatrix element in accordance with formula (22.5). The latter for shell lN is equal to... 

In these formulas the summation is over the coordinates of all electrons. Operators a 1) and L(1) in (22.18) compose a scalar product. However, L(1 acts only on This pecularity is indicated by the introduction of curly brackets. In order to find the matrix elements of operator (22.18), we have to transform it to the form... 

As in the non-relativistic case, matrix elements of operator (22.19) are calculated using first-order perturbation theory... 

In (22.37) ip is the wave function of an atom with motionless nucleus. The one-electronic submatrix element of the gradient operator (n/ V ni/i) is non-zero only for h = / 1. Therefore, the matrix element of operator (22.38) inside a shell of equivalent electrons vanishes and one has to account for this interaction only between shells. For the configuration, consisting of j closed and two open shells, it is defined by the following formula [156] ... 

In conclusion, we note that thus far we have derived matrix elements of the transformed Hamiltonian Xfor a given block in the complete matrix labelled by a particular value of rj rather than an effective Hamiltonian operating only within the subspace of the state rj. It is an easy matter to cast our results in the form of an effective Hamiltonian for any particular case since the matrix elements involved in either the commutator bracket formulation (contact transformation) or the explicit matrix element formulation (Van Vleck transformation) can always be factorised into a product of a matrix element of operators involved in X associated with the quantum number rj and a matrix element of operators that act only within the subspace levels of a given rj state, associated with the quantum number i. This follows because the basis set can be factorised as in equation (7.47). The matrix element involving the rj quantum number can then either be evaluated or included as a parameter to be determined experimentally, while the... 

Massieu function 48 mathematical constants 83, 90 mathematical functions 83 mathematical operators 84 mathematical symbols 81-86 matrices 83, 85 matrix element of operator 16 maxwell 115 Maxwell equations 123 mean free path 56 mean international ohm 114 mean international volt 114 mean ionic activity 58 mean ionic activity coefficient 58 mean ionic molality 58 mean life 22, 93 mean relative speed 56 mechanics classical 12 quantum 16 mega 74 melting 51 metre 70,71,110 micro 74 micron 110 mile 110 Miller indices 38 milli 74... 

Z Nuclear charge, exact Z Nuclear charge, reduced by the number of core electrons (n Bra n) Ket (n 0 m) Bracket (matrix element) of operator O between functions n and m (O) Average value of O ... 

Matrix elements of operators x andp can be evaluated by taking squares of Eqs (5.46). Some examples are given in the Exercises. 

We use a Fock space notation in terms of excitation operators and matrix elements of operators (using spin-orbitals) ... 

The expression above for the time correlation function can be read as a sequence of propagations and measurements. Reading from the right, the first propagator evolves the system from the nuclear configuration Rq and the electronic state a to configuration Rn and state /3 in a time t. At this time a measurement, i.e. the evaluation of a given matrix element of operator B,... 

A fundamental aspect of semi-empirical chemical bonding theories is their requirement that the model operators be state independent [56]. This property is, of course, not required of effective operators if only the numerical values are desired for the matrix elements of operators. Indeed, some semi-empirical theories, used in other areas of physics, do not impose the requirement of state independence. For instance, LS-dependent parameters are employed in describing the hyperfine coupling of two-electron atoms [31]. However, whenever effective operators themselves are the quantities of interest, as when studying semi-empirical theories of chemical bonding, state independence of effective operators becomes a necessity. This paper thus examines conditions leading to the generation of state-independent effective operators. 

The correspondence relations summarized in Section IIA show how any unitary transformation element can be evaluated within the classical limit. Sometimes, however, one is interested in matrix elements of operators which are not unitary. Consider, for example, the one-dimensional system discussed in the previous section if A is some operator, the question is how does one obtain the classical limit of the matrix element . 

Important examples of these expressions are the calculations of matrix elements of operators which are expanded by means of the spherical harmonic addition theorem (6, p. 141)... 

A major difficulty for molecular as opposed to atomic systems arises from the fact that two different reference axis systems are important, the molecule-fixed and the space-fixed system. Many perturbation related quantities require calculation of matrix elements of molecule-fixed components of angular momentum operators. Particular care is required with molecule-fixed matrix elements of operators that include an angular momentum operator associated with rotation of the molecule-fixed axis system relative to the space-fixed system. The molecule-fixed components of such operators have a physical meaning that is not intuitively obvious, as reflected by anomalous angular momentum commutation rules. 

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