Unitary orbital transformations

As proposed by Marzari and Vanderbilt [219], an intuitive solution to the problem of the non-uniqueness of the unitary transformed orbitals is to require that the total spread of the localized function should be minimal. The Marzari-Vanderbilt scheme is based on recent advances in the formulation of a theory of electronic polarization [220, 221]. By analyzing quantities such as changes in the spread (second moment) or the location of the center of charge of the MLWFs, it is possible to learn about the chemical nature of a given system. In particular the charge centers of the MLWFs are of interest, as they provide a classical correspondence to the location of an electron or electron pair. 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

Such a compact MCSCF wavefunction is designed to provide a good description of the set of strongly occupied spin-orbitals and of the CI amplitudes for CSFs in which only these spin-orbitals appear. It, of course, provides no information about the spin-orbitals that are not used to form the CSFs on which the MCSCF calculation is based. As a result, the MCSCF energy is invariant to a unitary transformation among these virtual orbitals. 

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

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. 

Normally the orbitals are real, and the unitary transformation becomes an orthogonal transformation. In the case of only two orbitals, the X matrix contains the rotation angle a, and the U matrix describes a 2 by 2 rotation. The connection between X and U is illustrated in Chapter 13 (Figure 13.2) and involves diagonalization of X (to give eigenvalues of ia), exponentiation (to give complex exponentials which may be witten as cos a i sin a), follow by backtransformation. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

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. 

With regard to the different points of view outlined in (a), (b) and (c), it should be pointed out that these differences arise mainly from the use of localized (a, LMO), or canonical (CMO, b, and c) molecular orbitals. In principle LMOs and CMOs are equivalent and are related by a unitary transformation. This can be illustrated by the C=C bonding in acetylene. 

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... 

Step 2. The set of CMOs orthogonal molecular orbitals (LMOs) Xj using, e.g., Ruedenberg s localization criterion205. This is achieved by multiplying up with an appropriate unitary transformation matrix L ... 

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... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

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... 

In each step the new orbital matrix Z r is evaluated through the unitary transformation Z r= Zr U. The MrxMr unitary matrix U is obtained solving the problem... 

It is worth noting that the methods mentioned give rise to Hartree-Fock equations based on different Fock operators for the various orbitals of the same spin. This implies that off-diagonal Lagrange multipliers appear which cannot be eliminated by a suitable unitary transformation. Moreover, the complications arise when excited states above the first one are considered (see, e.g. [25] for further details). 

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