Using Spatial Orbitals

In many applications, especially for programming purposes, it is very important to rewrite theoretical expressions containing spinorbitals in terms of spatial orbitals. This is imperative for the sake of effective computation, since the number of spinorbitals is twice as large as the number of spatial orbitals. If the computational procedure manipulates with the full list of two-electron integrals, the time requirement of the calculation is proportional at least to the fourth power of the number of basis functions. Consequently, if the programming would be done in terms of spinorbitals, this would lead to a 2 = 64-times longer run. The situation is usually worse, since the time requirement of a quantum chemical calculation including the approximate treatment of electron correlation is proportional to the (at least) 6th power of the number of orbitals. 

In this section we give some examples how one can rewrite spinorbital expressions in a simple manner. The basis of this transcription is that any integration over spinorbitals contains also summation over spin functions. 

Consider first the expectation value of a one-electron operator A  

The spin part can be either a and p. Formally, any summation over spinorbitals 

Using this formal identity, the expectation value in Eq. (9.1) becomes  

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Ansatz. The T s are written in terms of generators normal ordered with respect to a suitable closed shell vacuum 0). The first difference is operationally manifest as T s defined using spatial orbitals which do not commute with each other. The second difference arises due to our desire to have naturally truncating working equations after a finite power of T. This is accomplished by choosing to be of the following normal-ordered exponential form ... 

Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

Note that if using spatial orbitals the exchange integrals enter the energy expression with a factor of 1/2 relative to the Coulombic part. 

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

Open shell systems—for example, those with unequal numbers of spin up and spin down electrons—are usually modeled by a spin unrestricted model (which is the default for these systems in Gaussian). Restricted, closed shell calculations force each electron pair into a single spatial orbital, while open shell calculations use separate spatial orbitals for the spin up and spin down electrons (a and P respectively) ... 

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. 

In the HF scheme, the first origin of the correlation between electrons of antiparallel spins comes from the restriction that they are forced to occupy the same orbital (RHF scheme) and thus some of the same location in space. A simple way of taking into account the basic effects of the electronic correlation is to release the constraint of double occupation (UHF scheme = Unrestricted HF) and so use Different Orbitals for Different Spins (DODS scheme which is the European way of calling UHF). In this methodology, electrons with antiparallel spins are not found to doubly occupy the same orbital so that, in principle, they are not forced to coexist in the same spatial region as is the case in usual RHF wave functions. 

Sebastian has emphasized that (17a) implies Pi <0.5 (since 0restricted form the wavefunction (11) has when the a and / -spin orbitals are constrained to be equal. It can be circumvented by removing this constraint and using different spatial orbitals for electrons with different spin, which is accomplished by making different choices for the coupling functions. 

In this chapter, we later consider spin-polarized systems. One avenue of approach is to apply the spin unrestricted formalism, where SOs have different spatial orbitals for different spins. However, this procedure can introduce important spin contamination effects through the last term of Eq. (27) since the overlap matrix (5. These effects can be avoided by the use of spin-restricted theory. In this case only a single set of orbitals is used for a and / spins. 

This changes the way we write slater determinants. Using an overbar to denote 0 spin occupation of a spatial orbital, ,... 

The spatial orbitals the spin functions 0 and the coefficients of different if a linear combination of is used, may be explicitly determined by invoking the variational principle... 

The simplest and most widely used SCF procedure is the RHF, where the spatial orbitals are assumed as far as possible to be doubly occupied, and if there is molecular symmetry, to be of a pure symmetry type. As a... 

The variational condition determining the coefficients cJt is cubic in the unknowns, but iterative techniques permit these coefficients to be determined by repeated use of matrix diagonalization methods. Under most conditions it is possible to choose an iterative process facilitating convergence there is much RHF experience, and inordinate difficulties are not usually experienced. Because of the occupancy assumptions, it is possible without loss of generality to take the RHF spatial orbitals as orthogonal, and this is an important feature simplifying the calculations. 

One way to deal with unpaired electrons is to use the unrestricted HF (UHF). Whereas regular ab initio calculations restrict the one-electron spatial orbitals to be identical for a- and (3-spin electrons (so-called restricted HF, RHF), in UHF the orbitals are allowed to be different in the SCF processes. Usually, the difference in the spatial orbitals for a and (3 electrons is only slight. Unfortunately, when applied to a radical, UHF stumbles in a pitfall (97). It is called spin contamination. Unrestricted HF wave functions cannot be trusted to correspond to pure spin states such as a doublet for radicals or a singlet or triplet for diradicals. Theoretically speaking, the UHF wave function may not be an eigenfunction of the spin operators. 

The Slater wavefunction differs from the Hartree function not only in being composed of spin orbitals rather than just spatial orbitals, but also in the fact that it is not a simple product of one-electron functions, but rather a determinant (Section 4.3.3) whose elements are these functions. To construct a Slater wavefunction (Slater determinant) for a closed-shell species (the only kind we consider in any detail here), we use each of the occupied spatial orbitals to make two spin orbitals, by multiplying the spatial orbital by a and, separately, by jl. The spin orbitals are then filled with the available electrons. An example should make the procedure clear (Fig. 5.2). Suppose we wish to write a Slater determinant for a four-electron... 

The Slater determinant for the total wavefunction T of a 2 -electron atom or molecule is a 2 x 2 determinant with 2 rows due to the 2 electrons and 2 columns due to the 2 spin orbitals (you can interchange the row/column format) since these are closed-shell species, the number of spatial orbitals i// is half the number of electrons. We use the lowest n occupied spatial orbitals (the lowest 2n spin orbitals) to make the determinant. 

There are n spatial orbitals ij/ since we are considering a system of 2n electrons and each orbital holds two electrons. The 1 in parentheses on each orbital emphasizes that each of these n equations is a one-electron equation, dealing with the same electron (we could have used a 2 or a 3, etc.), i.e. the Fock operator (Eq. 5.36) is a one-electron operator, unlike the general electronic Hamiltonian operator of Eq. 5.15, which is a multi-electron operator (a 2n electron operator for our specific case). The Fock operator acts on a total of n spatial orbitals, the ij/1, Jj2,, i// in Eq. 5.35. 

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