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3. The Electronic Configurations, Term Symbols, and States Section treats the spatial, angular momentum, and spin symmetries of the many-electron wavefunctions that are formed as antisymmetrized products of atomic or molecular orbitals. Proper coupling of angular momenta (orbital and spin) is covered here, and atomic and molecular term symbols are treated. The need to include Configuration Interaction to achieve qualitatively correct descriptions of certain species' electronic structures is treated here. The role of the resultant Configuration Correlation Diagrams in the WoodwardHoffmann theory of chemical reactivity is also developed. 4. The Section on Molecular Rotation and Vibration provides an introduction to how vibrational and rotational energy levels and wavefunctions are expressed for diatomic, linear polyatomic, and non-linear polyatomic molecules whose electronic energies are described by a single potential energy surface. Rotations of "rigid" molecules and harmonic vibrations of uncoupled normal modes constitute the starting point of such treatments. 5. The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 6. The Section on More Quantitive Aspects of Electronic Structure C a l c u l a t i o n s introduces many of the computational chemistry methods that are used to quantitatively evaluate molecular orbital and configuration mixing amplitudes. The Hartree-Fock self-consistent field (SCF), configuration interaction (CI), multiconfigurational SCF (MCSCF), many-body and Møller-Plesset perturbation theories,...
10. Introduction to Quantum Mechanics, L. Pauling and E. B. Wilson, Dover Publications, Inc., New York, N. Y. (1963)- Pauling and Wilson. 11. Modern Quantum Chemistry, A. Szabo and N. S. Ostlund, Mc Graw-Hill, New York (1989)- Szabo and Ostlund. 12. Quantum Chemistry, I. N. Levine, Prentice Hall, Englewood Cliffs, N. J. (1991)Levine. 13. Energetic Principles of Chemical Reactions, J. Simons, Jones and Bartlett, Portola Valley, Calif. (1983),...
A. Operators Each physically measurable quantity has a corresponding operator. The eigenvalues of the operator tell the values of the corresponding physical property that can be observed...
where Ψ(qj,t) is the unknown wavefunction and H is the operator corresponding to the total energy physical property of the system. This operator is called the Hamiltonian and is formed, as stated above, by first writing down the classical mechanical expression for the total energy (kinetic plus potential) in cartesian coordinates and momenta and then replacing all classical momenta pj by their quantum mechanical operators pj = - ih∂/∂qj . For the H2O example used above, the classical mechanical energy of all thirteen particles is E = Σ i { pi2/2me + 1/2 Σ j e2/ri,j - Σ a Zae2/ri,a } + Σ a {pa2/2ma + 1/2 Σ b ZaZbe2/ra,b }, where the indices i and j are used to label the ten electrons whose thirty cartesian coordinates are {qi} and a and b label the three nuclei whose charges are denoted {Za}, and whose nine cartesian coordinates are {qa}. The electron and nuclear masses are denoted me and {ma}, respectively. The corresponding Hamiltonian operator is H = Σ i { - (h2/2me) ∂2/∂qi2 + 1/2 Σ j e2/ri,j - Σ a Zae2/ri,a } + Σ a { - (h2/2ma) ∂2/∂qa2+ 1/2 Σ b ZaZbe2/ra,b }. Notice that H is a second order differential operator in the space of the thirty-nine cartesian coordinates that describe the positions of the ten electrons and three nuclei. It is a second order operator because the momenta appear in the kinetic energy as pj2 and pa2, and the quantum mechanical operator for each momentum p = -ih ∂/∂q is of first order. The Schrödinger equation for the H2O example at hand then reads...
The number of dimensions depends on the number of particles and the number of spatial (and other) dimensions needed to characterize the position and motion of each particle 1. The Schrödinger Equation Consider an electron of mass m and charge e moving on a two-dimensional surface that defines the x,y plane (perhaps the electron is constrained to the surface of a solid by a potential that binds it tightly to a narrow region in the z-direction), and assume that the electron experiences a constant potential V0 at all points in this plane (on any real atomic or molecular surface, the electron would experience a potential that varies with position in a manner that reflects the periodic structure of the surface). The pertinent time independent Schrödinger equation is: - h2/2m (∂2/∂x2 +∂2/∂y2)ψ(x,y) +V 0ψ(x,y) = E ψ(x,y). Because there are no terms in this equation that couple motion in the x and y directions (e.g., no terms of the form xayb or ∂/∂x ∂/∂y or x∂/∂y), separation of variables can be used to write ψ as a product ψ(x,y)=A(x)B(y). Substitution of this form into the Schrödinger equation, followed by collecting together all x-dependent and all y-dependent terms, gives; - h2/2m A-1∂2A/∂x2 - h2/2m B-1∂2B/∂y2 =E-V0. Since the first term contains no y-dependence and the second contains no x-dependence, both must actually be constant (these two constants are denoted Ex and Ey, respectively), which allows two separate Schrödinger equations to be written: - h2/2m A-1∂2A/∂x2 =Ex, and - h2/2m B-1∂2B/∂y2 =Ey. The total energy E can then be expressed in terms of these separate energies Ex and Ey as Ex + Ey =E-V0. Solutions to the x- and y- Schrödinger equations are easily seen to be: A(x) = exp(ix(2mEx/h2)1/2) and exp(-ix(2mEx/h2)1/2) ,...
equations. An analogous process must be applied to B(y) to achieve a function that vanishes at y=0: B(y) = exp(iy(2mEy/h2)1/2) - exp(-iy(2mEy/h2)1/2) . Further requiring A(x) and B(y) to vanish, respectively, at x=Lx and y=Ly, gives equations that can be obeyed only if Ex and Ey assume particular values: exp(iLx(2mEx/h2)1/2) - exp(-iLx(2mEx/h2)1/2) = 0, and exp(iLy(2mEy/h2)1/2) - exp(-iLy(2mEy/h2)1/2) = 0. These equations are equivalent to sin(Lx(2mEx/h2)1/2) = sin(Ly(2mEy/h2)1/2) = 0. Knowing that sin(θ) vanishes at θ=nπ, for n=1,2,3,..., (although the sin(nπ) function vanishes for n=0, this function vanishes for all x or y, and is therefore unacceptable because it represents zero probability density at all points in space) one concludes that the energies Ex and Ey can assume only values that obey: Lx(2mEx/h2)1/2 =nxπ , Ly(2mEy/h2)1/2 =nyπ , or Ex = nx2π 2 h2/(2mLx2), and Ey = ny2π 2 h2/(2mLy2), with n x and ny =1,2,3, ... It is important to stress that it is the imposition of boundary conditions, expressing the fact that the electron is spatially constrained, that gives rise to quantized energies. In the absence of spatial confinement, or with confinement only at x =0 or Lx or only at y =0 or Ly, quantized energies would not be realized. In this example, confinement of the electron to a finite interval along both the x and y coordinates yields energies that are quantized along both axes. If the electron were confined along one coordinate (e.g., between 0 ≤ x ≤ Lx) but not along the other (i.e., B(y)...
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