N-electron wavefunction

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

Figure 4. Hierarchy of the SF models. Similar to the non-SF SR methods, the SF models converge to the exact n-electron wavefunction when the spin-flipping operator 0 includes up to n-tuple excitations. For example, the SF-CCSD model...

This is closely related to the possibility of writing the n-electron wavefunction as... 

Suppose that (xi,X2,..., xn) is the position-space representation of the N-electron wavefunction. It is a function of the space-spin coordinates Xk = (Ofcj ctjfc) in which is the position vector of the kth electron and Gk is its spin coordinate. The position-space wavefunction is obtained by solving the usual position- or r-space Schrodinger equation by one of the many well-developed approximate methods [32-34]. 

For the remainder of this Section, the primary focus is placed on forming proper N-electron wavefunctions by occupying the orbitals available to the system in a manner that guarantees that the resultant N-electron function is an eigenfunction of those operators that commute with the N-electron Hamiltonian. 

I. CSFs Are Used to Express the Full N-Electron Wavefunction... 

In such variational treatments of electronic structure, the N-electron wavefunction P is expanded as a sum over all CSFs that possess the desired spatial and spin symmetry ... 

The above expansion of the full N-electron wavefunction is termed a "configuration-interaction" (Cl) expansion. It is, in principle, a mathematically rigorous approach to expressing P because the set of all determinants that can be formed from a complete set of spin-orbitals can be shown to be complete. In practice, one is limited to the number of orbitals that can be used and in the number of CSFs that can be included in the Cl expansion. Nevertheless, the Cl expansion method forms the basis of the most commonly used techniques in quantum chemistry. 

The multiconfigurational self-consistent field (MCSCF) method in which the expectation value /is treated variationally and simultaneously made stationary with respect to variations in the Q and Cv,i coefficients subject to the constraints that the spin-orbitals and the full N-electron wavefunction remain normalized ... 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

In one sense, research in theoretical chemistry at Queen s University at Kingston originated outside the Department of Chemistry when A. John Coleman came in 1960 as head of the Department of Mathematics. Coleman took up Charles Coulson s challenge150 to make the use of reduced density matrices (RDM) a viable approach to the N-electron problem. RDMs had been introduced earlier by Husimi (1940), Lowdin (1955), and McWeeny (1955). The great attraction was that their use could reduce the 4N space-spin coordinates of the wavefunctions in the variational principle to only 16 such coordinates. But for the RDMs to be of value, one must first solve the celebrated N-repre-sentability problem formulated by Coleman, namely, that the RDMs employed must be derivable from an N-electron wavefunction.151 This constraint has since been a topic of much research at Queen s University, in the Departments of Chemistry and Mathematics as well as elsewhere. A number of workshops and conferences about RDMs have been held, including one in honor of John Coleman in 1985.152 Two chemists, Hans Kummer [Ph.D. Swiss Federal Technical... 

Assume that all n electron wavefunctions orthonormal—that is, that the overlap integrals involved in the secular equation Eq. (3.8.4) are replaced by Kronecker deltas. 

The most general n-electron wavefunction Oy can be written as a linear combination of the... 

The electronic charge density p(r) is a 3D function that can be calculated from the n-electron wavefunction solution T(r],r2,... Tp) of the electronic Schrbdinger equation of the molecule ... 

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],... 

An alternative way of dealing with the many-body problem is to bypass the calculation of the N-electron wavefunction by using the electron density p. The electron density can be defined as... 

Undoubtedly, the methods most widely used to solve the Schrodinger equation are those based on the approach originally proposed by Hartree [1] and Fock [2]. Hartree-Fock (HF) theory is the simplest of the ab initio or "first principles" quantum chemical theories, which are obtained directly from the Schrodinger equation without incorporating any empirical considerations. In the HF approximation, the n-electron wavefunction is built from a set of n independent one-electron spin orbitals which contain both spatial and spin components. The HF trial wavefunction is taken as a single Slater determinant of spin orbitals. 

Matrix elements between two n-electron wavefunctions can be factored into spin and spatial (including all information about orbital angular momenta) parts. It has been shown frequently that the Wigner-Eckart theorem can be applied as follows (Langhoff and Kern, 1977 McWeeny, 1965 Cooper and Musher, 1972) ... 

N-electron wavefunction P in this finite basis is given by ... 

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... 

Although Eqs. (1.2)-(1.5) contain all of the fundamental properties of the Fermion (electron) creation and annihilation operators, it may be useful to make a few additional remarks about how these operators are used in subsequent applications. In treating perturbative expansions of N-electron wavefunctions or when attempting to optimize the spin-orbitals appearing in such wavefunctions, it is often convenient to refer to Slater determinants that have been obtained from some reference determinant by replacing certain spin-orbitals by other spin orbitals. In terms of second-quantized operators, these spin-orbital replacements will be achieved by using the replacement operator as in Eq. (1.9). 

Methodology for the state-specific computation of N-electron wavefunctions of isolated states... 

The solution-processed films were annealed at 453 K under UHV for 1 h to further remove the contamination/solvent and improve the crystallinity. By comparing with the NEXAFS that has much deeper probing depth, we can get the clear picture of electronic states distribution from inside to outside. Overlap of n-electronic wavefunction within the lamellae planes should be responsible for the high intralayer mobility in P3HT films. On the other hand, Jt-electronic states distributed outside the polymer surface provide the possibility of forming overlap with the Jt-electronic wavefunction of an overlayer molecular material, consequently achieving efficient charge transfer in the related heterojunction structure used for polymer devices. 

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