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Introductory quantum chemistry(649s).pdf |
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2. The Section Simple Molecular Orbital Theory deals with atomic and molecular orbitals in a qualitative manner, including their symmetries, shapes, sizes, and energies. It introduces bonding, non-bonding, and antibonding orbitals, delocalized, hybrid, and Rydberg orbitals, and introduces Hückel-level models for the calculation of molecular orbitals as linear combinations of atomic orbitals (a more extensive treatment of...
coupled-cluster (CC), and density functional or Xα -like methods are included. The strengths and weaknesses of each of these techniques are discussed in some detail. Having mastered this section, the reader should be familiar with how potential energy hypersurfaces, molecular properties, forces on the individual atomic centers, and responses to externally applied fields or perturbations are evaluated on high speed computers....
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),...
has F= Σ j=1,N Zjexj, and F =Σ j=1,N Zjexj , where Zje is the charge on the jth particle. The mapping from F to F is straightforward only in terms of cartesian coordinates. To map a classical function F, given in terms of curvilinear coordinates (even if they are orthogonal), into its quantum operator is not at all straightforward. Interested readers are referred to Kemble's text on quantum mechanics which deals with this matter in detail. The mapping can always be done in terms of cartesian coordinates after which a transformation of the resulting coordinates and differential operators to a curvilinear system can be performed. The corresponding transformation of the kinetic energy operator to spherical coordinates is treated in detail in Appendix A. The text by EWK also covers this topic in considerable detail. The relationship of these quantum mechanical operators to experimental measurement will be made clear later in this chapter. For now, suffice it to say that these operators define equations whose solutions determine the values of the corresponding physical property that can be observed when a measurement is carried out; only the values so determined can be observed. This should suggest the origins of quantum mechanics' prediction that some measurements will produce discrete or quantized values of certain variables (e.g., energy, angular momentum, etc.). B. Wavefunctions The eigenfunctions of a quantum mechanical operator depend on the coordinates upon which the operator acts; these functions are called wavefunctions In addition to operators corresponding to each physically measurable quantity, quantum mechanics describes the state of the system in terms of a wavefunction Ψ that is a function of the coordinates {qj} and of time t. The function |Ψ(qj,t)|2 = Ψ*Ψ gives the probability density for observing the coordinates at the values qj at time t. For a manyparticle system such as the H2O molecule, the wavefunction depends on many coordinates. For the H2O example, it depends on the x, y, and z (or r,θ, and φ) coordinates of the ten...
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...
If the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrödinger equation In cases where the classical energy, and hence the quantum Hamiltonian, do not contain terms that are explicitly time dependent (e.g., interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms discussed later in this text), the separations of variables techniques can be used to reduce the Schrödinger equation to a time-independent equation. In such cases, H is not explicitly time dependent, so one can assume that Ψ(qj,t) is of the form Ψ (qj,t) = Ψ(qj) F(t). Substituting this 'ansatz' into the time-dependent Schrödinger equation gives Ψ (qj) i h ∂F/∂t = H Ψ(qj) F(t) . Dividing by Ψ(qj) F(t) then gives F-1 (i h ∂F/∂t) = Ψ -1 (H Ψ(qj) ). Since F(t) is only a function of time t, and Ψ(qj) is only a function of the spatial coordinates {qj}, and because the left hand and right hand sides must be equal for all values of t and of {qj}, both the left and right hand sides must equal a constant. If this constant is called E, the two equations that are embodied in this separated Schrödinger equation read as follows: H Ψ(qj) = E Ψ(qj) , i h ∂F(t)/∂t = ih dF(t)/dt = E F(t). The first of these equations is called the time-independent Schrödinger equation; it is a so-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues of H....
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