Cartesian derivatives

C SHAPE FUNCTIONS THEIR CARTESIAN DERIVATIVES ARE READ FROM A WORK FILE C... 

This expansion can be continued beyond the terms which are second order in deviations of from (i), but only at the cost of calculating the corresponding higher order ah initio Cartesian derivatives (which is not sufficiently efficient at present57). 

If the form (2.120) of Tf t is used with the Cartesian derivatives (2.121), then the double contraction of two transition tensors for the states i and / of normal mode p becomes ... 

The spherical form of the multipole expansion is very useful if we are looking for the explicit orientational dependence of the interaction energy. However, in some applications the use the conceptually simpler Cartesian form of the operators V1a 1b may be more convenient. Moreover, unlike the spherical derivation, the Cartesian derivation is very simple, and can be followed by everybody who knows how to differentiate a function of x, y and z 149. To express the operator V,, in terms of Cartesian tensors we have to define the reducible, with respect to SO(3), tensorial components of multipole moments,... 

Derivatives of the energy and molecular properties are most conveniently calculated in Cartesian coordinates. Due to translational and rotational symmetries these Cartesian derivatives are not independent. For example, for forces only 3N — m (m = 5, 6) Cartesian components are independent. The remaining five or six components can be calculated by taking into consideration the translational and rotational symmetries of the energy. This has two useful applications. First, the number of derivatives that must be calculated ab initio may be reduced. This saving is especially important for small molecules. Alternatively, one may calculate all derivatives ab initio and simply use translational and rotational symmetries as a simple check on the calculation. 

To obtain the formula for V in cylindrical coordinates we employ the definition of the V-operator in Cartesian coordinates (C.57), eliminate the Cartesian unit vectors by (C.66) and eliminate the Cartesian derivative operators by (C.64). The resulting formula for the V operator in cylindrical coordinates can then be used to calculate all the necessary differential operators in cylindrical coordinates provided that the spatial derivatives of the unit vectors er,eg,ez are used to differentiate the unit vectors on which V operates. 

U. DinurandA. T. Hagler,/ Chem. PAys., 91,2949 (1989), Determination of Atomic Point Charges and Point Dipoles from the Cartesian Derivatives of the Molecular Dipole Moment and Second Moments, and from Energy Derivatives of Planar Dimers. I. Theory. 

The spherical tensor gradient operator (STGO) is a solid harmonic whose Cartesian coordinate arguments have been replaced by Cartesian derivatives. Hobson s theorem [25] is the result of acting a STGO upon any spherical function /(r )... 

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