Slater determinantal wave function

Note that this property is a correlation effect unique to electrons of the same spin. If we consider the contrasting Slater determinantal wave function formed from different spins... 

If we carry out a restricted HF calculation, one or other of the degenerate frontier pair will be chosen to be occupied, the calculation will optimize the shapes of all of the occupied orbitals, and we will end up with a best possible single-Slater-determinantal wave function formed from those MOs. But it should be fairly obvious that an equally good wave function... 

The wavefunction F° then follows as an antisymmetrized product built from the single-particle functions q>i(r, ms) for the Z electrons (Slater determinantal wave-function, see below and Section 7.2), where r is the spatial vector and ms the spin magnetic quantum number. 

Let s assume that, for instance, after the Hartree-Fock SCF computation is undertaken the set of canonical molecular one-electronic orbitals are determined so that the Slater determinantal wave function for 2N... 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

We shall be concerned with ground and excited electronic states which can be adequately described by a single determinantal wave function, i.e. doublet states, triplet states, etc. with spin 5/0). Let 0 be the Slater determinant constructed from a set of spin-orbitals consisting of spatial part ( = 1,2,. ..,n") associated with a spin functions and orbitals... 

Density functional theory, 21, 31, 245-246 B3LYP functional, 246 Hartree-Fock-Slater exchange, 246 Kohn-Sham equations, 245 local density approximation, 246 nonlocal corrections, 246 Density matrix, 232 Determinantal wave function, 23 Dewar benzene, 290 from acetylene + cyclobutadiene, 290 interaction diagram, 297 rearrangement to benzene, 290, 296-297 DFT, see Density functional theory... 

Slater determinant, 251, see also Determinantal wave function... 

Ts is defined as the expectation value of the kinetic-energy operator T with the Slater determinant arising from density n, i.e., Ts[n] = ( [n] T kinetic energy is defined as T[n = ( I,[n] Tj I,[n]. All consequences of antisymmetrization (i.e., exchange) are described by employing a determinantal wave function in defining Ts. Hence, Tc, the difference between Ts and T is a pure correlation effect. 

A consequence of the antisymmetry property of the determinantal wave functions is that the matrix elements can be evaluated according to the Slater rules. Before the application of the Slater rules the determinantal functions should be ordered in such a way that maximum coincidence among the positions of the spinorbitals is secured. Notice that each transposition in the position of spinorbitals alters the sign of the determinantal function. 

In the molecular orbital method the ground-state determinantal wave function f o) = f (0,0 0) is constructed from the set of occupied molecular spin-orbitals cpi).Then tne matrix elements of the one- and two-electron operators can be evaluated by utilising the Slater rules. The expectation value of a one-electron operator is... 

Slater determinant, 251, see also Determinantal wave function SnI mechanism, 129-130 alkyl halides, 129 carbocation intermediates in, 106 leaving group, 130 Lewis acid catalysis, 130 Sn2 mechanism, 130-136 alkyl halides, 130 carbocation intermediates in, 106 and E2, 143 gas phase, 144 geometry of approach, 131 leaving group, 130, 132 nucleophilicity, 131-132 substituent effects, 132-134 transition state, 132, 133 VBCM description, 134-135 Snoutene, 247 rearrangement, 289 Sodimn borohydride, 83, 278 Sodimn hydride, 83 Soft Electrophiles, 110 reaction with etiolate, 110 Spin function, 234... 

This is known as a Slater determinant or a determinantal wave function. Since... 

In discussing SCF calculations, we used an orthogonalized 2s AO of the form a(fr) + b 2s), where 2s is a (nodeless) 2s Slater-type orbital. Tliis procedure is justified by use of the freedom of adding a multiple of one column of a determinant to another the determinantal wave function for the oxygen atom is the same whether it is the 2s or the orthogonalized 2s AO that is used.]... 

We have seen that a Hartree product is truly an independent-electron wave function since the simultaneous probability of finding electron-one in d i at x, electron-two in d 2 at X2, etc. is simply equal to the product of the probabilities that electron-one is in dX, electron-two is in dx2, etc. Antisymmetrizing a Hartree product to obtain a Slater determinant introduces exchange effects, so-called because they arise from the requirement that be invariant to the exchange of the space and spin coordinates of any two electrons. In particular, a Slater determinant incorporates exchange correlation, which means that the motion of two electrons with parallel spins is correlated. Since the motion of electrons with opposite spins remains uncorrelated, it is customary to refer to a single determinantal wave function as an uncorrelated wave function. 

Slater, J. C., Quantum Theory of Matter, 2nd ed., McGraw-Hill, New York, 1968. Chapter 11 discusses determinantal wave functions and derives expressions for matrix elements between determinants in a somewhat different way than we have done. 

To calculate expectation values, we have to get rid of the determinantal wave function in matrix elements over a many-electron operator O such as the many-electron Hamiltonian and reduce them to computable one-electron and two-electron matrix elements. This can be done easily using creators and annihilators since we can expand a Slater determinant according to Eq. (8.127)... 

Among possible approaches, the so-called second quantization plays an important role. The ultimate goal of the second quantized approach to the many-electron problem is to offer a formalism which is substantially simpler than the traditional one in many cases. As a matter of fact, most difficulties of the traditional or first quantized approach arises from the Pauli principle which requires the wave function W of Eq. (1.1) to be antisymmetric in the electronic variables. This is an additional requirement which does not result from the Schrodinger equation and requires a special formalism the using of Slater determinants for constructing appropriate solutions to Eq. (1.1). The Slater determinant is not a very pictorial mathematical entity, and the evaluation of matrix elements over determinantal wave functions makes the first quantized quantum chemistry somewhat complicated for beginners. In the second quantized... 

Matrix elements between determinantal wave functions can be evaluated by the so-called Slater (or Slater-Condon) rules. We shall not derive the Slater rules in general, but in some particular cases the first and second quantization-based derivations will be compared. 

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