21_580ln_fa08

21_580ln_fa08 - MIT OpenCourseWare http://ocw.mit.edu 5.80...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture #21: Construction of Potential Curves by the Rydberg-Klein-Rees Method (RKR) Go to http: // leroy.uwaterloo.ca / programs.html for RKR1 [computes V ( r ) by RKR from G ( v ) and B ( v )] and LEVEL [numerical integration to get v ( r ) and compute many v | O p | v integrals]. The RKR method is an accurate and convenient method of constructing a potential curve, V ( r ) from observed vibrational energy levels and rotational constants. To use the method one need only know or guess the variation of the vibrational energy and rotational constant with v , the vibrational quantum number. The RKR method generates inner and outer turning points ( r and r + , the two points on the potential curve corresponding to a specific energy) by numerical evaluation of two integrals involving the G ( v ) and B ( v ) functions. The potential curve is generated point-wise. The validity of the RKR method rests on an approximate criterion for the existence of a vibra- tional eigenvalue at an energy E . This criterion is known as the Bohr-Sommerfeld quantization condition which can be derived by the WKB (Wentzel, Kramers, Brillouin) method in which the wave function is represented as a truncated asymptotic expansion. In what is called the WKB approximation or the semi-classical approximation the WKB wavefunction has the form r p ( r ) 1 / 2 i = exp r p ( r ) dr (1) where p ( r ), the classical mechanical momentum at r , is by definition p ( r ) = 2 ( E V ( r )) 1 / 2 . (2) This expression for will be valid wherever the de Broglie wavelength is neither infinite nor rapidly varying with r . ( r ) = = (3) p ( r ) [2 ( E V ( r ))] 1 / 2 | ( r ) | < implies r is far from a turning point. d 1 implies that dV ( r ) is small, or that there are no dr dr kinks in the potential. We can summarize these requirements as dV dr 1 ....
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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21_580ln_fa08 - MIT OpenCourseWare http://ocw.mit.edu 5.80...

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