Wavefunction spatial

As noted above, many of the common molecular properties don t depend on electron spin. The first step is to average-out the effect of electron spin, and we do this by integrating with respect to si and S2 to give the purely spatial wavefunction... 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

Another approach to the problem of rare gas scattering is to replace the spatial wavefunctions of Eq. (11.4) with their Fourier transforms, the momentum space wavefunctions. These wavefunctions represent the velocity distributions of the electron in the Rydberg states. Proceeding along these lines, we rewrite Eq. (11.4)... 

We construct the molecular product states which have the spatial wavefunctions... 

If we now imagine that AE varies in time, but slowly, its only effect is to cause a time variation of the energy of the n state. We assume that the spatial wavefunction is unaffected by AE and that no transitions occur. This approximation is the adiabatic approximation of Autler and Townes.11 Now let us consider the time variation of AE of particular interest to us, Emw cos cot. If the assumptions stated above are valid, we can use the energy of Eq. (15.6) as the unperturbed Hamiltonian H0 in the Schroedinger equation. Explicitly,... 

The eigenfunctions of the atom must be eigenfunctions of the total angular momentum and its projection on a space fixed z axis. If we ignore the electron spins for simplicity, the total angular momentum L and its projection M on the space fixed z axis are conserved. The spatial wavefunctions iM(r,R) are related to the body fixed wavefunctions by the Euler transformation... 

Symmetry arguments tell us that we have to take a = b, so we write possible spatial wavefunctions as... 

In the Bom interpretation (Section 4.2.6) the square of a one-electron wavefunction ij/ at any point X is the probability density (with units of volume-1) for the wavefunction at that point, and j/ 2dxdydz is the probability (a pure number) at any moment of finding the electron in an infinitesimal volume dxdydz around the point (the probability of finding the electron at a mathematical point is zero). For a multielectron wavefunction T the relationship between the wavefunction T and the electron density p is more complicated, being the number of electrons in the molecule times the sum over all their spins of the integral of the square of the molecular wavefunction integrated over the coordinates of all but one of the electrons (Section 5.5.4.5, AIM discussion). It can be shown [9] that p(x, y, z) is related to the component one-electron spatial wavefunctions ij/t (the molecular orbitals) of a single-determinant wavefunction T (recall from Section 5.2.3.1 that the Hartree-Fock T can be approximated as a Slater determinant of spin orbitals i/qoc and i// /i) by... 

When one of the electrons in helium is excited from the Is to the 2s orbital, the configuration Is 2s is obtained. Taking the indistinguishibility of electrons into account, two spatial wavefunctions can be written ... 

With increasing atomic volume, one approaches the free atom limit where Hund s first rule postulates maximum spin, so that the individual spins of the electrons in a shell are aligned parallel. More generally, Pauli s exclusion principle implies that electrons with parallel spins have different spatial wavefunctions, reduces the Coulomb repulsion and is seen as exchange interaction. When the atoms are squeezed into a solid, some of the electrons are forced into common spatial wavefunctions, with antiparallel spins and reduction of the overall magnetic moment. At surfaces and interfaces, the reduced coordination reverses this effect, and a part of the atomic moment is recovered. 

When r-i = r2 the triplet product functions, Eqs. (3.7.4)-(3.7.6), vanish The two electrons cannot be localized in the same spot this is called a "Fermi hole." When r1 = rz, then the singlet spatial wavefunction, Eq. (3.7.3), is finite and nonzero. This consequence of Fermi-Dirac statistics is called "spin pairing" Two electrons with opposite spins attract each other (despite the classical Coulomb repulsion), while two electrons with the same spin repel each other. 

The m-th sideband state is displaced from the carrier, or the ( , 3) state in the absence of microwaves, by rtm. Note that the spatial wavefunction of Eq. (9) is unchanged. 

We close this section by noting that the destructive interference has its origin in the spatial wavefunction, which gives cancellations in the scattering matrix element if the amplitudes for particle a (or j3) has opposite signs when the neutron scatters from sites 1 and 2. Fig. 5 is an attempt to illustrate the scattering from a particle that is delocalized over two different sites. 

Figure 1.13 Spatial wavefunction of a 2px AO. The shading (filled/empty) denotes the change in sign of the wavefunction. The probability (2px)2dv is >0 throughout...

We have m xm equations because each of the m spatial MO s xjr we used (recall that there is one HF equation for each Eqs (5.47)) is expanded with m basis functions. The Roothaan-Hall equations connect the basis functions 0 (contained in the integrals F and S, Eqs (5.55)), the coefficients c, and the MO energy levels e. Given a basis setf i, 5 = 1, 2, 3,. .., m] they can be used to calculate the c s, and thus the MOs ir (Eq. (5.52)) and the MO energy levels e. The overall electron distribution in the molecule can be calculated from the total wavefunction iR, which can be written as a Slater determinant of the component spatial wavefunctions i/r (by including spin functions), and in principle anyway, any property of a molecule can be calculated... 

Table 3.4 lists the spin functions, each of which appears along the diagonal of one determinant constructed from unpaired molecular spatial wavefunctions. For example, if there are two half-filled subshells, a( 1) and b(2), using Table 3.4, the singlet wavefunction is written as a sum of two determinants ... 

As mentioned in the Introduction, magnetic exchange is both electrostatic and quantum mechanical in nature. It is electrostatic because the relevant energies are related to the energy costs of overlapping electron densities. It is quantum mechanical because of the fundamental requirement that the total wavefunction of two electrons must be antisymmetric to the exchange of both the spin and spatial coordinates of the two electrons. The wavefunction is separable into a product of spatial wavefunction f r, ri) that is a function of the positions r and ri of the two electrons, and a spin coordinate wavefunction /(cri, crz), where a, is the Pauli matrix for the spin operator Si = haijl. Both i/r (ri, r2) and xCcti, 02) can be symmetric or antisymmetric individually but the fundamental... 

We consider systems containing IV 4 particles, ofrespective masses m,, with a spatial wavefunction depending on their positions Fi, ..., Fjv in a space-fixed coordinate system. These coordinates can be transformed into the position R of the system s center of mass and relative coordinates Fjjv,. ..,Fjv where r,-, stands for r — r we also define r,y = lr,yl and f,y = r,y/r,y. We restrict our attention to isolated systems in which it is not necessary to consider the motion of the center of mass, so the wavefunction can be expressed entirely in terms of the 31V — 3 relative coordinates, ofwhich three (two if IV = 2) can be chosen to specify the orientation of the entire wavefunction with respect to space-fixed axes, while the remaining coordinates can describe the internal stmcture of the wavefunction. The restriction to IV 4 causes the number of internal coordinates to be equal to the number of distinct r. Considering also that orbital angular momentum must be conserved, the two-particle system can have the trivial spatial wavefunction... 

The square of the wavefunction, for a given configuration (orbital occupation), is interpreted as the electron density at that point in space. Since the electronic Hamiltonian does not explicitly have a term for spin, our wavefunction does not either. To completely describe the electron distribution the spin coordinate, i, must be included. The spin coordinate takes on two values -I- V2 and — V2. The spin wavefunction for spin aligned along the positive z-axis is a(C), and the spin wavefunction for spin aligned along the negative z-axis is P( ). The product of the spatial wavefunction (atomic orbital) < )(r) and the spin wavefunction a( ) or P( ) is the complete wavefunaion and is called a spin-orbital, denoted by x(f, Q. For an -electron system, the simplest wavefunction would be in the form of a product of spin-orbitals. 

Table 1 illustrates the six wavefunctions that are available in ATOMPLUS along with sample output from solving the Schrodinger equation with these trial wavefunctions for the He atom. Table 1 depicts only the spatial part of the wavefunction. Each spatial wavefunction is symmetric the total wavefunction is the product of this spatial part and the spin part which is antisymmetric. 

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