3502_36_dece11_LG - Chem 3502/5502 Physical Chemistry II...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 36, December 11, 2009 (Some material in this lecture has been adapted from Cramer, C. J. Essentials of Computational Chemistry , Wiley, Chichester: 2002; pp. 191-203.) Solved Homework Consider first, for an individual vibration U vib = R h " k B e h " / k B T # 1 ( ) If we replace the exponential by its power series expansion, truncating after the first power in ω (since ω is going to zero, and thus only the first power term has significant size compared to the others) we have U vib = R h " k B 1 + h " k B T # 1 $ % & ( ) = RT So, there appears to be no problem with this term blowing up as ω goes to zero. Now consider S vib = R h " k B T e h " / k B T # 1 ( ) # ln 1 # e # h " / k B T ( ) $ % & & ( ) ) The first term in brackets is just the one we did above times 1/ T , so we know that it converges to 1 as ω goes to zero. What about the second term? Using the power series expansion, we have R ln 1 " e " h # / k B T ( ) = R ln 1 " 1 + h # k B T $ % & ( ) = R ln # + R ln h k B T $ % & ( )
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36-2 The second term on the r.h.s. is just a constant, but the first term goes to negative infinity as ω goes to zero. This is actually correct. The entropy of a harmonic oscillator becomes infinite as the vibrational frequency goes to zero (implying that there is really no well anymore; thus, you just have a free particle, and since a free particle has a continuum of states open to it, it has infinite entropy given an infinite volume). However, as the frequency goes to zero (meaning a very “soft” mode in the molecule), the harmonic oscillator is a lousy approximation for the potential energy surface. Typically such modes are torsions (rotations about single bonds), and as such they don’t look like parabolic wells on the PES, they are instead threefold periodic potentials that repeat after one full rotation. The partition function for an internal rotor is different from that for a harmonic oscillator, and does not diverge at zero frequency (a so-called “free rotor”). Electron Correlation Multiconfiguration Self-Consistent Field Theory. Hartree-Fock theory makes the fundamental approximation that each electron moves in the static electric field created by all of the other electrons, and then proceeds to optimize orbitals for all of the electrons in a self-consistent fashion subject to a variational constraint. The resulting wave function, when operated upon by the Hamiltonian, delivers as its expectation value the lowest possible energy for a single-determinantal wave function formed from the chosen basis set . So, the question arises of how we might modify the HF wave function to obtain a lower electronic energy when we operate on that modified wave function with the Hamiltonian. By the variational principle, such a construction would be a more accurate wave function. We cannot do better than the HF wave function with a single determinant, so one obvious choice is to construct a wave function as a linear combination of multiple determinants, i.e., " = c 0 " HF + c 1 " 1 + c 2 " 2 + L (36-1) where the coefficients c
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Spring '08 term at Minnesota.

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3502_36_dece11_LG - Chem 3502/5502 Physical Chemistry II...

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