Symmetric wave function, definition

The definitions are here given under the assumption that the wave function XP is either antisymmetric or symmetric for a trial function without symmetry property, one has to replace the binomial factor NCV before the integrand by a factor l/p and sum over the N(N—l). . . (N—p+l) possible integrals which are obtained by placing the fixed coordinates x, x 2,. . ., x P in various ways in the N places of the first factor W and the fixed coordinates xv x2,. . xv similarly in the second factor W. By using Eq. II.8 we then obtain... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

Problems associated with the quantum-mechanical definition of molecular shape do not diminish the importance of molecular conformation as a chemically meaningful concept. To find the balanced perspective it is necessary to know that the same wave function that describes an isolated molecule, also describes the chemically equivalent molecule, closely confined. The distinction arises from different sets of boundary conditions. The spherically symmetrical solutions of the free molecule are no longer physically acceptable solutions for the confined molecule. 

By definition, the Hamiltonian of a system of identical particles is invariant under the interchange of all the coordinates of any two particles. The wave function describing the system must be either symmetric or antisymmetric under this interchange. If the particles have integer spin, the wavefunction is symmetric and the particles are called bosons if they have half-integer spins, the wavefunction is antisymmetric and the particles are fermions. Our discussion will be restricted to electrons, which are fermions. 

Pauli exclusion principle follows mathematically from definition of wave function for a system of identical particles - it can be either symmetric or antisymmetric (depending on particles spin). 

The wave function is antisymmetric with respect to the exchange of the coordinates of any two electrons, and, therefore p is symmetric with respect to such an exchange. Hence, the definition of p is independent of the label of the electron we do not integrate over. According to this definition. 

With the li orbital and one of the 2p orbitals we can make wave functions analogous to those from the (1 i)(2i) configuration. We use the complex functions so that we have definite values of m. With the li space orbital and the 2pl space orbital we have a symmetric space factor... 

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