Antisymmetry conditions

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

In Equation 6.16, K represent certain electronic configurations, and usually have the form of so-called Slater determinants that are in accord with the Pauli principle and the antisymmetry condition (the sign of K changes if the coordinates of two electrons are interchanged see Atkins [1983], Jensen [1998], Klessinger [1982], Levine [1991], Springborg [2000]). These //-electron Slater determinants are built from orbitals as one-electron functions t, the latter of which are multi-center molecular orbitals in the case of molecular orbital schemes (as opposed to, for example, valence-bond schemes where the one-electron functions are much more localized). 

For an electronic wavefunction, antisymmetry is a physical requirement following from the fact that electrons are fermions. It is essentially a requirement that y agree with the results of experimental physics. More specifically, this requirement means that any valid wavefunction must satisfy the following condition ... 

The fundamental laws which determine the behavior of an electronic system are the Schrodinger equation (Eq. II. 1) and the Pauli exclusion principle expressed in the form of the antisymmetry requirement (Eq. II.2). We note that even the latter auxiliary condition introduces a certain correlation between the movements of the electrons. 

It should be emphasized that not all normalizable hermitean matrices r(x x 2. . . x xlx2. . . xp) having the correct antisymmetry property are necessarily strict density matrices, i.e., are derivable from a wave function W. For instance, for p — N, it is a necessary and sufficient condition that the matrix JT is idempotent, so that r2 = r, Tr (JH) = 1. This means that the F-space goes conceptually outside the -space, which it fully contains. The relation IV. 5 has apparently a meaning within the entire jT-space, independent of whether T is connected with a wave function or not. The question is only which restrictions one has to impose on r in order to secure the validity of the inequality... 

The orbital phase continuity conditions stem from the intrinsic property of electrons. Electrons are fermions, and are described by wavefnnctions antisymmetric (change plus and minus signs) with respect to an interchange of the coordinates of an pair of particles. The antisymmetry principle is a more fnndamental principle than Pauli s exclusion principle. Slater determinants are antisymmetric, which is why the overlap integral between t(a c) given above has a negative... 

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

Thus the singlet spatial function is symmetric and the triplet one antisymmetric. If we use the variation theorem to obtain an approximate solution to the ESE requiring symmetry as a subsidiary condition, we are dealing with the singlet state for two electrons. Alternatively, antisymmetry, as a subsidiary condition, yields the triplet state. 

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

Because of antisymmetry, variations of that are simply Unear transformations of occupied orbitals have no effect other than a change of normalization. For orbital functions with fixed normalization, a general variation of takes the form = Hi ni Ha( 1 — na) fScf. The variational condition is... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

The ambiguity is removed by requiring that all terms (—l) (x tp + (k)) of (7.26) be identical. This is a stronger condition than overall antisymmetry. 

Cases of three or more electrons were very difficult to treat by the above methods. For instance, for three-electron systems, it is required to have six terms in the expansion of each basis function in order to comply with the antisymmetry criterion, and each term must have factors containing ri2, ri3, r23, etc., if we want to accelerate the convergence. There is, indeed, a real problem with the size of each trial wave function. A symmetrical wavefunction requires that the trial basis set for helium contain two terms to guarantee the permutation of electrons. For an N-electron system, this number grows as N . For a ten-electron system like water, it would be required that each basis set member have more than 3 million terms, and this is in addition to the dependence on 3N variables of each of the terms. These conditions make the Schrodinger equation intractable for systems of even a few electrons. Just the bookkeeping of these terms is practically impossible. 

When determinants are used, the antisymmetry requirement does actually, surprisingly, impose a degree of conditional probability on the distribution of electrons with like spin, a point which will be relevant later on. 

Valency forces are also electrostatic in nature. A consistent application of the quantum laws to two hydrogen atoms shows that the pair may exist in a lower energy level than the isolated individuals, but only on condition that their electrons have opposite spins. This condition is imposed by the requirement of antisymmetry in the wave function. Analogous conclusions apply to other atoms, and the limitations on the possible electron states which the Pauli principle demands restrict combination to the satiuation of specific valencies. Valency forces fall off exponentially as distance between atoms increases. 

A much more drastic sanction resides in the requirement that the total wave function of the system shall be antisymmetrical, because in certain circumstances this imposes on the electrons the condition of parallel spins. In the hydrogen molecule the electrons possess opposite spins in virtue not of any mutual force which they exert directly upon one another, but of the condition that the lowest energy state demands an orbital wave function of the symmetrical type. The categorical requirement of overall antisymmetry then dictates the antiparallel spin function. According to the calculation of Heisenberg, there are certain crystals for which the lowest energy state involves the parallel alignment of the spins. These crystals are... 

Due to the antisymmetry requirement for the wave function (see Chapter 1), the holes have to satisfy some general (integral) conditions. The electrons with parallel spins have to avoid each other fh r, r2)Ar2 =... 

The forerunner of Cl is the self-consistent field (SCF) method [1, 2]. A version that properly accounts for the antisymmetry of the electronic wave function was developed independently by Fock [3] and Slater [4] shortly after Schrodinger s papers. It is characterized by an approximate wave function that is a single determinant whose elements are one-electron functions (spin orbitals). The latter orbitals are optimized under two conditions minimization of the energy expectation value and mutual orthonormality. The method produces both the occupied orbitals appearing in the determinant but also a potentially infinite number of unoccupied functions that prove to be the basis for the Cl method. One can look upon a Slater determinant formed by substituting unoccupied for occupied one-electron functions as a representation of an excited state of the molecular system. The possible applications to spectroscopy were obvious. 

Due to the antisymmetry requirement for the wave function (see Chapter 1), the holes have to satisfy some general (integral) conditions. The electrons with parallel spins have to avoid each other //j (ri,r2)dr2 = / xc (r, T2)dr2 = —1 (one electron disappears from the neighborhood of the other), while the electrons with opposite spins are not influenced by the Pauli exclusion principle /hxc (ri, r2)dr2 = /hxc (fi, r2)dr2 = 0. The exchange correlation hole is a sum of the exchange hole and the correlation hole hj" = -I- hg, ... 

Here r, represents the cartesian coordinates of the i-th electron, and x, the quasi-coordinates that are modified by the backflow displacement vector which is dependent on all coordinates r. The quasi-coordinates are constructed subject to the condition that the antisymmetry of any wave function is retained. The idea of backflow in a quantum system goes back to... 

By the above considerations we have seen that the antisymmetry of the Hartree-Fock wave function results in a transformational degree of freedom for the MOs. As it was discussed previously, this is consistent with the condition Fp = 0 which is the expanded form of the Brillouin theorem. The latter has a particular significance also in post-Hartree-Fock calculations. This property will be utilized in the next section. 

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