Constraining force constant determination

Here there are four measured frequencies with which to constrain three independent force constants, so the best-fitting force constants can be determined through an iterative least squares fit, minimizing S(v eas-Vcaic) - Assuming average atomic masses of 51.996 and 15.9994 for chromium and oxygen, respectively, the best-fit force constants are Ai = 495.2 Newtons/m,iT = 21.3 Newtons/m, andFjv = 44.7 Newtons/m. These force constants show the typical relationship K H,Fj j,. Calculated frequencies are ... 

Problem 7-8. Consider the case of a heteronuclear diatomic molecule constrained to move in one dimension. Let the masses of the nuclei be denoted by m and M, and the force constant by k. Set up and solve the secular equation determine that the allowed modes of motion are the overall translation and vibration. Determine the vibrational frequency in terms of m, M and k. 

The refinement calculation may be carried out in a variety of ways, and a few general remarks should be made before we consider particular examples. We wish to determine re, /2, /3, and /4, where these denote symbolically the equilibrium structure (which may be thought of as the linear force field), the quadratic, cubic, and quartic force field. (Terms higher than quartic are not considered here.) Each set of data depends on all constants up to a certain order, as shown in Table 3 for example, Ae, Be, and Ce depend only on re, the oo values depend on re and /2, the a values on re, /2, and /3, and the x values on re, /2, /3, and /4. Ideally one should refine all data simultaneously to all force constants (including the equilibrium structure), but in practice the calculation has to be broken down into steps. Thus usually the equilibrium structure re, or some approximation to re, is determined first from the rotational constants then the quadratic force field /2 is determined from the o>, , and r values holding re constrained then the cubic force field /, is determined from the values holding re and /2 constrained and finally the quartic force field /, is determined from the x values holding re, /2, and/3 constrained. (This should be compared with the discussion for diatomic molecules at the end of Section 3.)... 

Table 10 shows the force fields of H20, H2S, and H2Se as determined by Strey and Mills,60 with 3 quartic interaction constants constrained to zero. This calculation was refined to to, a, and x values, the quadratic force field ... 

To determine the free-energy profile (rather then the potential energy profile as above), one performs a (typically short) constant temperature (canonical ensemble) FPMD simulation of the system (while constraining the reaction coordinate to the prescribed value) and records the mean force of constraint along the chosen reaction coordinate. This process is then repeated for a number of points along the reaction coordinate connecting the reactant and... 

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