Del2
Bob this is an old derivation of the Kinetic
energy for a function of
. Note the first derivative divided by r. This is the part of del2 that I
miss if I assume d=x. I what follows,
this derivation will “essentially” be repeated for the molecular system.
Consider a function of r.
Numerical Partials with respect to Z 
Kolos Coordinates --- picture at #H2 in Kolos’ coordinates
Change variables to the center of mass coordinate system.
(2.1)
Define the wave function as
These were defined in the Reviews of Modern Physics article[1] --
see ..\nonad\Note of interest.htm. The wave function does not depend explicitly
on ,
but rather implicitly through the center of mass position and on
(2.3)
The del2 with respect to the nuclear coordinates is
(2.4)
The first derivative with respect to ZA begins as
The partial of D is
The partials of zk1 and zk2 are(2.7)
Thus the first derivative is
(2.9)
The second derivative is
(2.10)
When summed over x and y, the third term -1/D while the second term
3/D giving rise to the familiar 2/D of
equation (1.1). The others become
(2.11)
The expectation value of the Kinetic energy is
(2.12)
H2 in Kolos’ coordinates
At large values of |D|, the wave function is centered around the nuclei so that
r1 and r2 are both approximately equal to D/2
(2.13)
This wave function form depends explicitly on D in the large D limit, so that it is not surprising that a large part of the Kolos nuclear kinetic energy comes from the partials with respect to D, even in this limit.
H2 in Alexander/Coldwell coordinates
Figure 2
The wave function form is
These coordinates can be written in terms of the Kolos coordinates
Note that there are more terms than in Kolos’ wave function. The Kolos variables for the electron coordinates are a simple origin shift, so that derivatives with respect to them are equivalent to derivatives with respect to the original variables. That is not true of the coordinates in (3.2).
The first derivative of the wave function with respect to ZA is similar to that given by (2.5).
The change comes in equation (2.8).
(3.4)
The derivative of r1A with
respect to ZA is -1, not -,
with the same true of the derivative with respect to r2A while the derivatives with respect to r1B and r2B are both zero.
This leads to
(3.5)
The second derivative is
(3.6)
Summing over X, Y and Z for nucleus A produces
(3.7)
Then summing over the two nuclei gives
(3.8)
The expectation value of the Kinetic energy is
The wiggle term, the one that exists even when the partials with respect to D go to zero, is the term in { }.
To make an explicit connection with Kolos’ terms, these need to be put in terms of partials with respect to k1 and k2. The relations in (3.2) allow this to be done. The first derivatives are of the form
(3.10)
The seconds are
(3.11)
The cross term is
The A B terms in (3.12) are not in equation (3.9). In terms of the Kolos derivatives the first, second and fifth terms in (3.9) are equal to del2 k1 while the third, fourth and sixth term are equal to del2 k2. Thus
The symmetry of the system with respect to electron 1 and 2, allows a single del2 to count for both so that
This is twice the heading of one of Kolos’ columns The second one in ..\nonad\KolosTABLE III.htm. In the large D limit, this accounts for all of Enuclear. The Kolos cross term is not present.
Where do the cross terms go?
In this section use (3.2) to define the wave function as
(4.1)
The first derivatives involve the dependence of z1A, z1B, z2A, and z2B on z both directly and indirectly through zk1 and zk2
(4.2)
The derivatives of the coordinates are either
1 or leading to
(4.3)
Then finally
(4.4)
This does agree with (3.3) owing to the cancellation of the derivatives in the 2 terms involving B. This comes about due to the fact that instead of the wave function form in equation (2.2), we have used a form with more dependence on RA and RB. In particular as shown by (3.2) the from in use involves
(4.5)
with this last vector just right to make (3.14) the nuclear kinetic energy.