Slater matrix

With the various orbital-types described, one can construct a simple many-electron wave function by forming the antisymmetrized product of orbitals. This is conveniently evaluated using the determinant of the Slater matrix, whose elements are the values of each orbital (in rows) evaluated at the coordinates of each electron (in columns). 

The Slater matrix for an Ai-electron determinant will contain elements consisting of the N occupied orbitals evaluated at each electron s coordinates. If the molecular orbitals are expanded using a linear combination of basis functions (i.e. )i(x) = and the number of basis functions is 0(N) then O(N )... 

Linear scaling can be achieved when localized molecular orbitals (LMOs) are used to create sparsity within the Slater matrix. The density of a LMO is confined to limited region of space around its centroid, so only a few LMOs need to be evaluated for each electron. The second consideration on the way to linear scaling is to accelerate the transformation from basis function to LMO s. Linear scaling QMC methods have been a popular field of research and several algorithms have already been published. 

The MO evaluation algorithm devised by Aspuru-Guzik et al. [163] truncated the orbital transformation using a numerical cutoff rather than a spatial one. Their algorithm uses a 3-D grid to identify sparsity in the Slater matrix. For each point on the grid, the threshold Cfj,iX (x)>lO is applied to create a list of relevant... 

The //yj matrices are, in practice, evaluated in temis of one- and two-electron integrals over the MOs using the Slater-Condon mles [M] or their equivalent. Prior to fomiing the Ffjj matrix elements, the one-and two-electron integrals. 

The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... 

Inter-atomic two-centre matrix elements (cp the hopping of electrons from one site to another. They can be described [7] as linear combmations of so-called Slater-Koster elements [9], The coefficients depend only on the orientation of the atoms / and m. in the crystal. For elementary metals described with s, p, and d basis fiinctions there are ten independent Slater-Koster elements. In the traditional fonnulation, the orientation is neglected and the two-centre elements depend only on the distance between the atoms [6]. (In several models [6,... 

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater Is orbital with a Gaussian function. That is, we replace the Is radial wave function... 

For any sizeable system the Slater determinant can be tedious to write out, let alone the equivalent full orbital expansion, and so it is common to use a shorthand notation. Various notation systems have been devised. In one system the terms along the diagonal of the matrix are written as a single-row determinant. For the 3x3 determinant we therefore have ... 

The orbitals used for methane, for example, are four Is Slater orbitals of hydrogen and one 2s and three 2p Slater orbitals of carbon, leading to an 8 x 8 secular matrix. Slater orbitals are systematic approximations to atomic orbitals that are widely used in computer applications. We will investigate Slater orbitals in more detail in later chapters. 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions... 

For first- and seeond-row atoms, the Is or (2s, 2p) or (3s,3p, 3d) valenee-state ionization energies (aj s), the number of valenee eleetrons ( Elee.) as well as the orbital exponents (es, ep and ej) of Slater-type orbitals used to ealeulate the overlap matrix elements Sp y eorresponding are given below. 

The anti symmetrized orbital produet A (l)i(l)2Cl)3 is represented by the short hand (1>1(1>2(1>3 I and is referred to as a Slater determinant. The origin of this notation ean be made elear by noting that (1/VN ) times the determinant of a matrix whose rows are labeled by the index i of the spin-orbital (jii and whose eolumns are labeled by the index j of the eleetron at rj is equal to the above funetion A (l)i(l)2Cl)3 = (1/V3 ) det(( )i (rj)). The general strueture of sueh Slater determinants is illustrated below ... 

One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N-Electron Configuration Eunctions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon Rules Provide this Capability... 

II. The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSFs... 

To form the Hk,l matrix, one uses the so-ealled Slater-Condon rules whieh express all non-vanishing determinental matrix elements involving either one- or two- eleetron operators (one-eleetron operators are additive and appear as... 

The Slater-Condon rules give the matrix elements between two determinants... 

As a first step in applying these rules, one must examine > and > and determine by how many (if any) spin-orbitals > and > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to aehieve maximal eoineidenee with those in the other determinant it is essential to keep traek of the number of permutations ( Np) that one makes in aehieving maximal eoineidenee. The results of the Slater-Condon rules given below are then multiplied by (-l) p to obtain the matrix elements between the original > and >. The final result does not depend on whether one ehooses to permute ... 

Onee maximal eoineidenee has been aehieved, the Slater-Condon (SC) rules provide the following preseriptions for evaluating the matrix elements of any operator F + G eontaining a one-eleetron part F = Zi f(i) and a two-eleetron part G = Zij g(i,j) (the Hamiltonian is, of eourse, a speeifie example of sueh an operator the eleetrie dipole... 

The full N terms that arise in the N-eleetron Slater determinants do not have to be treated explieitly, nor do the N (N + l)/2 Hamiltonian matrix elements among the N terms of one Slater determinant and the N terms of the same or another Slater determinant. 

In this form, it is elear that E is a quadratie funetion of the Cl amplitudes Cj it is a quartie funetional of the spin-orbitals beeause the Slater-Condon rules express eaeh <

Cl matrix element in terms of one- and two-eleetron integrals < > and... 

The Slater-Condon rules allow one to express the Hamiltonian matrix elements appearing here as... 

The amplitude for the so-ealled referenee CSF used in the SCF proeess is taken as unity and the other CSFs amplitudes are determined, relative to this one, by Rayleigh-Sehrodinger perturbation theory using the full N-eleetron Hamiltonian minus the sum of Foek operators H-H as the perturbation. The Slater-Condon rules are used for evaluating matrix elements of (H-H ) among these CSFs. The essential features of the MPPT/MBPT approaeh are deseribed in the following artieles J. A. Pople, R. Krishnan, H. B. Sehlegel, and J. S. Binkley, Int. J. Quantum Chem. 14, 545 (1978) R. J. Bartlett and D. M. Silver, J. Chem. Phys. 3258 (1975) R. Krishnan and J. A. Pople, Int. J. Quantum Chem. 

As a result, the exaet CC equations are quartic equations for the ti , ti gte. amplitudes. Although it is a rather formidable task to evaluate all of the eommutator matrix elements appearing in the above CC equations, it ean be and has been done (the referenees given above to Purvis and Bartlett are espeeially relevant in this eontext). The result is to express eaeh sueh matrix element, via the Slater-Condon rules, in terms of one- and two-eleetron integrals over the spin-orbitals used in determining , ineluding those in itself and the Virtual orbitals not in . 

The first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

In my discussion of pyridine, I took a combination of these determinants that had the correct singlet spin symmetry (that is, the combination that represented a singlet state). I could equally well have concentrated on the triplet states. In modem Cl calculations, we simply use all the raw Slater determinants. Such single determinants by themselves are not necessarily spin eigenfunctions, but provided we include them all we will get correct spin eigenfunctions on diago-nalization of the Hamiltonian matrix. 

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. 

---