Unitary transformation notation

We now prove that any unitary transformation of the orbitals leaves a closed-shell SCF Slater-determinant wave function unchanged, thereby showing the validity of transforming to localized MOs. Let i//loc be the SCF wave function written using localized orbitals. We wish to prove that if/loc equals can, where pc n uses the canonical MOs. In the notation of (1.260), we have... 

Warnings (i) The Tpq do not form a second-rank tensor and so unitary transformations must be carried out using the four-index notation Tijki. (ii) The contraction of TiJki may be accompanied by the introduction of numerical factors, for example when 7(4) is the elastic stiffness (Nye (1957)). 

Dirac Notation for Integrals Energy Functional to Be Minimized Energy Minimization with Constraints Slater Determinant Subject to a Unitary Transformation The J and K Operators Are Invariant Diagonalization of the Lagrange Multipliers... 

The j(qA ) form a set of independent coordinates if the u lk) are such a set since they are related by the unitary transformation defined by (2.11). We note, however, as a new feature, that the <(qA ) are complex. Both the kinetic energy T [from (2.14)] and the potential energy O [from (2.18)] are hermitian and therefore the Lagrangian L derived from these forms is hermitian as required. In tensor notation we obtain... 

If exact decoupling is algorithmically achieved by only one unitary transformation as in the case of X2C, we shall call it a one-step transformation. However, this notation may be ambiguous as the single unitary transformation can be decomposed into or constructed from more than one unitary transformation. 

C is then referred to as the matrix that brings to diagonal form both M and H it describes a new basis in which the operator H has vanishing matrix elements between different functions. When only orthonormal sets are admitted, M = 1, and C is the matrix of a unitary transformation as defined in (2.2.24) such that = C . In the notation used previously... 

Consider a Turing computable function f(i) that maps the positive integers IN onto a subspace of IN. Then we know from the previous section that there is a quantum Turing machine based on a reversible, classical machine M on which this function can be evaluated. The overall computation of / is described by the unitary operator Uf which is the product of local, unitary transformations Ui. To abbreviate the notation, we will only consider the subspace of the input and output data of the quantum machine. Furthermore, we will write i) to denote a part of the memory in which the number i is stored. For example, using the binary number system,... 

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