Scalar operator

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

The matrix elements (8.35) in the uncoupled space-fixed basis can be most easily evaluated if all interaction operators are represented as uncoupled products of spherical tensors, with each tensor defined in the space-fixed coordinate system. Since the Hamiltonian is always a scalar operator, we can write any interaction in the Hamiltonian as a sum... 

By contrast, any n-dimensional vector space has a generic way, by means of a basis set, to be mapped onto 72 , such that the basic operations becomes simply the corresponding element-by-element scalar operations on the arrays. Furthermore, the basis set can always be choosen such that the scalar product assumes the standard form... 

In order to find expressions for relativistic matrix elements of the energy operator we have to utilize the following formula for two-particle scalar operators ... 

In many physical problems we come across excited configurations consisting of several open shells or at least one electron above the open shell. Therefore, we have to be able to transform wave functions and matrix elements from one coupling scheme to the other for such complex configurations. If K denotes the configuration, and m, fifi stand for the quantum numbers of two different coupling schemes, then for the corresponding wave functions formulas of the kind (12.1), (12.2) hold, whereas the matrix element of some scalar operator D transforms as ... 

Zero-rank tensor operators (scalar operators), K = 0 ... 

The simplest example is a scalar operator 2Tq0-1 with the transformation property... 

This means that scalar operators are invariant with respect to rotations in coordinate or spin space. An example for a scalar operator is the Elamiltonian, i.e., the operator of the energy. 

We are mainly interested in compound tensor operators of rank zero (i.e., scalar operators such as the Hamiltonian). To form a scalar from two tensor operators 0 and /l, their ranks k and j have to be equal. Further, the +q component of lk> has to be combined with the -q component of and... 

The dipolar spin-spin coupling operators are scalar operators of the form //1 . A 2 y.11. The tensorial structure of JCss becomes apparent if we write the Breit-Pauli spin-spin coupling operator as... 

It is possible to add and subtract matrices using + and —, but remember to ensure that the two (or more) mattices have the same dimensions as has the destination. It is also possible to mix matrix and scalar operations, so that the syntax =2 MMULT(X,Y) is acceptable. Furthermore, it is possible to add (or subtract) matrices consisting of a constant number, for example =Y +2 would add 2 to each element of Y. Other conventions for mixing matrix and scalar variables and operations can be determined by practice, although in most cases the result is what we would logically expect. 

The main characteristics of the CRAY-1 computer cure shown in Table I (see also reference (2 ). The scalar operations are seen to be approximately twice that of the CDC 7600 and IBM 360/195. 

Although GATHER is a scalar operation and rather slow, 12 machine cycles/element, (a machine cycle is 12.5 ns),... 

The Newton-Raphson method uses information obtained by taking the first and second derivatives of the energy wilh respect to the coordinates. - The combination of both firsi and second derivatives provides a powerful method to locale minima. This may be a time-consuming process becau.se of the matrix manipulations that must be undertaken for a 3N system, where N is the number of atoms. In Equation 28-2. >. V is the dot product of V multiplied by itself. Note that V is a scalar operator. 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

Before the details of the implementation of MCSCF methods are discussed, it is useful to introduce a few general computer programming concepts. Modern computers may be classified in several ways depending on size, cost, capabilities, or architecture. One such classification divides computers into scalar and vector machines. Scalar computers (e.g. the VAX 11/780) perform primitive arithmetic operations such as additions and multiplications on pairs of arguments. Vector computers (e.g. the CRAY X-MP and CYBER-205) have, in addition to scalar operations, vector instructions ... 

Since A is a 4-vector field and is a scalar operator, it follows that 4p is a (polar)vector field. Let us now choose the (axial) pseudovector field Bp that accompanies 4p so that 1) it satisfies the same Lorentz gauge as 4 M, that is, oMBp = 0, and it solves the field equation (that accompanies (6) for 4p ... 

Since a is a scalar operator, even this simple energy-dependent elimination of the small component permits an exact separation of the spin-free and spin-dependent terms of the Dirac Hamiltonian by applying Dirac s relation... 

Energy, which is the observable quantity associated with the Hamiltonian operator, is a pure number or, more precisely, a scalar quantity. Therefore, the Hamiltonian must be a scalar operator. In this section, the prescriptions for constructing a scalar operator from combinations of more complicated operators, such as vector angular momenta, and for evaluating matrix elements of these composite scalar operators are reviewed briefly. 

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