Unitary Transformations of Orbitals

By now expanding both of the above exponential operators, we obtain 

The exponential matrix exp(iA) appearing in Eq. (1.44) is defined through the power series appearing in that equation. However, as we show below, this matrix can be computed from the A matrix in a much more straightforward and practical manner. 

If we want the transformation described by exp(iA) to preserve orthonormality of the spin-orbitals or, equivalently, to preserve the anticommutation relations [see discussion following Eq. (1.5)] 

Now if the operator A is required to be hermitian, which then makes the elements A form a hermitian matrix 

When the matrix A is hermitian, it can be divided into real and imaginary parts 

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The perturbation selection breaks the invariance of the SAC/SAC-Cl energies on the unitary transformation of orbitals among the occupied and/or unoccupied manifolds. This is the source of the discontinuity in the potential energy surface, if the external perturbation induces a large (sudden) mixing of orbitals within the occupied and unoccupied orbitals. Suppose a deformation of benzene from the Dgh geometry to less symmetric one, the degenerate set of orbitals in occupied or unoccupied manifold makes a sudden (discontinuous) deformation of MOs (see Ref. [44] for some details]. The MOD method [46] can solve this problem that may occur in the optimization... 

After a suitable unitary transformation of orbitals, analogous to what we have done in the case of GHF. 

Finally we note that self-interaction-corrected (SIC) methods and the GW approximation cannot be treated within the introduced generalized exchange interaction because these methods employ a different operator for each orbital, leading to a non-orthogonalized set of orbitals (see also Section 3.7.2). Moreover the SIC energy-functional is not invariant under a unitary transformation of orbitals. ... 

Note that in Eqs. (143, 144) we didn t specify which set of orbitals (HF or KS) are used, because Eqs. (143, 144) are invariant with respect to unitary transformation of orbitals. The eigenvalues and Ef differ... 

Miich is invariant (like R itself) under a unitary transformation of orbitals— and then write R as a sum of orbital contributions (R = Cj c ), we find... 

The X matrix contains the parameters describing the unitary transformation of the M orbitals, being of the size of M x M. The orthogonality is incorporated by requiring that the X matrix is antisymmetric, = —x , i.e. 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

The inclusion of (nonrelativistic) property operators, in combination with relativistic approximation schemes, bears some complications known as the picture-change error (PCE) [67,190,191] as it completely neglects the unitary transformation of that property operator from the original Dirac to the Schrodinger picture. Such PCEs are especially large for properties where the inner (core) part of the valence orbital is probed, for example, nuclear electric field gradients (EEG), which are an important... 

The localized molecular orbitals (LMOs) can be defined as the unitary transformation of CMOs that (roughly speaking) makes the transformed functions as much like the localized NBOs as possible,24... 

Remember that the square of the wave function, or any of the reduced density matrices, are independent of a unitary transformation of the orbitals. Hence, any pair of orbitals is as good as the other. However, the chemical picture of molecular orbitals is easily understood for most of the chemists. In this case, it is easier looking... 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

The localized many-body perturbation theory (LMBPT) applies localized HF orbitals which are unitary transforms of the canonical ones in the diagrammatic many-body perturbation theory. The method was elaborated on models of cyclic polyenes in the Pariser-Parr-Pople (PPP) approximation. These systems are considered as not well localized so they are suitable to study the importance of non local effects. The description of LMBPT follows the main points as it was first published in 1984 (Kapuy etal, 1983). 

The 1-RDM can be diagonalized by a unitary transformation of the spin orbitals (jf), (x) with the eigenvectors being the natural spin orbitals (NSOs) and the eigenvalues , representing the ONs of the latter. 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

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... 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

In this section we shall go through some of the formalism needed for the coming derivation of the optimization methods. The parameters to be varied in the energy expression (3 25) are the Cl coefficients and the molecular orbitals. We will consider these variations as rotations in an orthonormalized vector space. For example, variations of the MO s correspond to a unitary transformation of the original MO s into a new set ... 

So far we have considered an arbitrary unitary transformation of the spin-orbitals. In practice we shall only transform the spatial part of the orbitals, the... 

According to the general relationship (5.9), rotations in isospin space transform the electron creation operators by the D-matrix of rank 1/2. If we go over from these operators to the one-electron wave functions they produce, then we shall have the unitary transformation of radial orbitals... 

We wish to extend our acknowledgements to Professor Ahlrichs for reading the preprint. In his opinion our discussion on pages 44—46 is too conclusive with respect to the justifiability of CEP A. He suggests that the arguments on p. 46 should be so reformulated that it is possible to assume such an unitary transformation of occupied and virtual orbitals which would conform best to the approximations inherent in Eqs. (170) and (172). 

An infinitesimal unitary transformation of the orbital basis that modifies must mix occupied and unoccupied orbital functions. For a typical orbital variation, <5 = a8c . Unitarity induces 8a = —functional derivatives 8/8

[Pg.60]

All these statements, although correct in principle, are not precise from the technical point of view. For example, the zero approximate wave function in the PCILO method is a one-electron approximate function constructed from the bond wave functions determined by an a posteriori localization procedure from an HFR function. Thus the bond orbitals appear after a unitary transformation of the canonical MOs, which correspond to some more or less arbitrary localization criteria [123-125]. 

The following identity [4-6,20,21], which indicates the invariance of a Slater determinantal wavefunction to a unitary transformation of a pair of occupied Ms = +1/2 spin or Ms = -1/2 spin orbitals, will be used on numerous occasions in subsequent sections of this Chapter. 

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