Subject: Re: CCL:force constants of diatomics in GAUSSIAN-03

From: frisch.-at-.gaussian.com (Michael Frisch)
Date: Wed, 21 Jul 2004 10:24:03 -0400

On Wed, Jul 21, 2004 at 03:15:48PM +0400, Dmitry Rozmanov wrote:
Dmitry Rozmanov wrote
> If this is the case, then I guess this is just a wrong way of doing things 
> and the force constants got by Gaussian are not correct at all. There is a
> definition of the thing and there is no two way of calculation.
> 
> ---Dmitry.
> 

This is nonsense.  The details are in a white paper on our web site,
but the key point is that the force constant is the second derivative
with respect to a normal mode displacement and the units for the
normal mode, or equivalently the convention for what consititues a
unit step, are arbitrary.

For polyatomic molecules, one typically diagonalizes the force
constant matrix in mass-weighted coordinates, so the natural unit step
is a normalized displacement in these coordinates.  This approach is
general and applicable to any polyatomic molecule.  In the particular
case of H2 with the molecule along the x-axis, this normalized step
would be (1/sqrt(2),0,0,1/sqrt(2),0,0) in the 6 cartesian coordinates.
This unit step changes the H-H distance by sqrt(2).  For the particular
case of diatomic molecules when people calculate by hand, they use the
distance between atoms as the coordinate, which simpler for diatomics
but doesn't apply to polyatomics.  In that coordinate system, a unit
displacement changes the distance by 1 rather than sqrt(2), so the force
constants (second derivatives of the energy) differ by a factor of 2.
The corresponding reduced masses for the mode also differ by a factor
of two and the frequency is the same.  

For the diatomic, the "by hand" coordinates give a reduced mass for
the mode which is the same as the overall reduced mass for the
molecule.  For a general polyatomic molecule, the reduced mass
corresponding to a particular mode is not an observable quantity and
is not defined until one adopts a convention for the (arbitrary) size
of a unit normal mode displacement.

Mike Frisch