Constraining force constant

Standard errors in brackets in the last digits quoted indicates a constrained force constant. 

A single harmonic oscillator constrained to the a -axis has one force constant k that—stretching a point—we might think of A as a 1 x 1 force constant matr ix. Two oscillators that interact with one another lead to a 2 x 2 force constant matr ix... 

The penalty function approach adds a tenn of tire type k r — ro) to the function to be optimized. The variable r is constrained to be near the target value ro, and the force constant k describes how important the constraint is compared with the unconstrained optimization. By making k arbitrary large, tire constraint may be fulfilled to any given... 

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. 

Following our hypothesis that bradykinin adopts a C-terminal (3 turn upon complexation with the receptor, the , / backbone dihedral angles in the tetrapeptide corresponding to the C-terminus of bradykinin (Ser-Pro-Phe-Agr) were constrained in a harmonic fashion (force constant = 15 Kcal A-1 mof ) to values that define a type If [3-turn [43]. This tetrapeptide probe was then systematically translated about the interior of a theoretical box inscribing the rat... 

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.)... 

For HCN the situation is somewhat better, because the data on DCN are much more effectively independent of the HCN data. This molecule has also been the subject of much high-resolution spectroscopic study, so that the vibration-rotation energy levels are particularly well known and its vibrational spectrum is free of accidental resonances. Table 8 compares the results of three quite different calculations. The calculation by Strey and Mills is the most recent, and was based on the latest spectroscopic data the refinement was made to a and x values rather than to the vibrational levels and rotational constants as used by both the earlier workers. Strey and Mills also constrained 3 of the quartic interaction constants to zero, and refined to cubic and quartic force constants in a separate calculation to the quadratic refinement. The level of agreement between the calculations leads to conclusions rather similar to those made above for C02 in particular, standard errors should be multiplied by at... 

Standard errors are indicated in brackets after each force constant. Note that these quartic force constants were constrained to zero. 

Certain force constants may be constrained to zero, particularly when they represent interactions between remote coordinates in the... 

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