Quantum chemistry Slater determinants

The primary challenge in quantum chemistry is to find a good approximation to the electronic wave function of a quantum state. We can express any N-electron wave function in a complete basis of Slater determinants, through the FCI expansion,... 

Based on first principles. Used for rigorous quantum chemistry, i. e., for MO calculations based on Slater determinants. Generally, the Schrodinger equation (Hy/ = Ey/) is solved in the BO approximation (see Born-Oppenheimer approximation) with a large but finite basis set of atomic orbitals (for example, STO-3G, Hartree-Fock with configuration interaction). 

One of the most important concepts of quantum chemistry is the Slater determinant. Most quantum chemical treatments are made just over Slater determinants. Nevertheless, in many problems the formulation over Slater determinants is not very convenient and the derivation of final expressions is very complicated. The advantage of second quantization lies in the fact that this technique permits us to arrive at the same expressions in a considerably simpler way. In second quantization a Slater determinant is represented by a product of creation and annihilation operators. As will be shown below, the Hamiltonian can also be expressed by creation and annihilation operators and thus the eigenvalue problem is reduced to the manipulation of creation and annihilation operators. This manipulation can be done diagrammatically (according to certain rules which will be specified later) and from the diagrams formed one can write down the final mathematical expression. In the traditional way a Slater determinant I ) is specified by one-electron functions as follows ... 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

The most uniformly successful family of methods begins with the simplest possible n-electron wavefunction satisfying the Pauli antisymmetry principle - a Slater determinant [2] of one-electron functions % r.to) called spinorbitals. Each spinorbital is a product of a molecular orbital xpt(r) and a spinfunction a(to) or P(co). The V /.(r) are found by the self-consistent-field (SCF) procedure introduced [3] into quantum chemistry by Hartree. The Hartree-Fock (HF) [4] and Kohn-Sham density functional (KS) [5,6] theories are both of this type, as are their many simplified variants [7-16],... 

The self-consistent field Hartree-Fock (HF) method is the foundation of AI quantum chemistry. In this simplest of approaches, the /-electron ground state function T fxj,. X/y) is approximated by a single Slater determinant built from antisymmetrized products of one-electron functions i/r (x) (molecular orbitals, MOs, X includes space, r, and spin, a, = 1/2 variables). MOs are orthonormal single electron wavefunctions commonly expressed as linear combinations of atom-centered basis functions ip as i/z (x) = c/ii /J(x). The MO expansion coefficients are... 

We begin our discussion of wave function based quantum chemistry by introducing the concepts of -electron and one-electron expansions. First, in Sec. 2.1, we consider the expansion of the approximate wave function in Slater determinants of spin orbitals. Next, we introduce in Sec. 2.2 the one-electron Gaussian functions (basis functions) in terms of which the molecular spin orbitals are usually constructed the standard basis sets of Gaussian functions are finally briefly reviewed in Sec. 2.3. 

The Hartree-Fock and the Kohn-Sham Slater determinants are not identical, since they are composed of different single-particle orbitals, and thus the definition of exchange and correlation energy in DFT and in conventional quantum chemistry is slightly different [52]. 

The simplest trial function employed in ab initio quantum chemistry is the single Slater determinant fimction in which N spin orbitals are occupied by N electrons ... 

Computational Quantum Chemistry the Slater determinant, becomes... 

In the vast majority of the quantum chemistry literature, Slater determinants have been used to express antisymmetric N-electron wavefunctions, and explicit dilTerential and multiplicative operators have been used to write the electronic Hamiltonian. More recently, it has become quite common to express the operators and state vectors that arise in considering stationary electronic states of atoms and molecules (within the Born-Oppenheimer approximation) in the so-called second quantization notation (Linderberg and Ohrn, 1973). The electron creation ) and annihilation... 

The second intrusion of the electron spin came through a non-energetic, symmetry requirement, the so-called Eermi-Dirac statistics for systems of identical, half-integer spin particles, which results in total antisymmetry of the Schrbdinger wave function in a combined space and spin coordinate domain. This entails the Pauli exclusion principle (1925) in the framework of the independent-particle, Slater-determinantal model. The expression of atomic and molecular wave functions as iinear combinations of Slater determinants has been the basis of most of the subsequent methodologies of quantum chemistry, thermodynamics, and spectroscopy. 

Of course, the answer is no. The absolute limit M is equal to half the number of the eleetrons. beeause only then ean we create N spinorbitals and write the Slater determinant. However, in quantum chemistry (rather misleadingly), we call the minimum basis set the basis set resulting from an inner shell and valence oibitals in the corresponding atoms. For example, the minimum basis set for a water molecule is I5,2s and three 2p orbitals of oxygen and two I5 orbitals of hydrogen atoms, which is seven AOs in total (while the truly minimal basis would contain only 10/2 = 5 AOs). 

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