Duo is a general program for solving a coupled rovibronic problem for diatomics.
+ +Within the Born-Oppenheimer approximation the spin-orbit (SO) free
+Hamiltonian of the diatomic problem in the absence of the hyperfine splitting
+ is given by 2:
![](download/temp/LaTeX2692729341814108318.png)
where is the electronic part,
is the rotational part and
is the vibrational part.
+The rotational angular momentum is perpendicular to the internuclear axis. Considering Hund's case a and representing the rotational angular momentum as
![](download/temp/LaTeX6769203689468309757.png)
in terms of the total , electronic
, and spin
angular momenta and choosing the body-fixed
axis along the internuclear axis the rotational part is given by
![](download/temp/LaTeX4985196685252994968.png)
where ,
and
are the corresponding momentum ladder operators.
The duo basis set functions are chosen as the product
+![](download/temp/LaTeX8164832238879656507.png)
where is the electronic function,
is the rotational function,
is the vibrational function; 'state' is the label identifying the electronic state;
,
, and
are the
axis projections of the electronic, spin and total angular momenta, respectively, and
;
is the projection of the total angular momentum along the laboratory axis
;
is the vibrational quantum number.
For the rotational basis set the rigid rotor functions are used, with the following matrix elements:
+![](download/temp/LaTeX2957261370005360606.png)
where is omitted for the simplicity.
The electronic basis functions appear in the solution only implicitly as matrix elements of different components of the operator as well as of the spin-orbit contributions. We choose these basis functions to satisfy the following conditions^2^:
![](download/temp/LaTeX5307144792322887259.png)
where is a reflection through the molecular-fixed
-plane (parity operator) and
for
states and
for all other states. The following non-vanishing matrix elements of the spin operators are valid:
![](download/temp/LaTeX6457220849845493965.png)
where the label 'states' is omitted for the simplicity.
+ +For example, the non-vanishing and symmetrically unique matrix elements of the ladder angular momentum operators as well as of the SO-matrix elements on this basis functions for the three lowest states of AlO
,
and
are obtained using MOLPRO and can be summarized as follows:
![](download/temp/LaTeX5836138745718601059.png)
where is omitted for the simplicity.
+All other non-vanishing matrix elements can be obtained by applying the symmetric properties upon the reflection (see above).
It should be noted for the MOLPRO functions are not eigenfunctions of
as required by the equation above. For example, the electronic state
gives rise to two degenerate solutions labelled in MOLPRO as
and
. The corresponding matrix representation of the
operator is not diagonal as computed by MOLPRO with the matrix elements is usually given by
![](download/temp/LaTeX8778477990539368986.png)
Besides the matrix elements of and
as well as the spin-orbit matrix elements are complex numbers in the MOLPRO representation.
+The transformation from and
(MOLPRO) to our representation
(duo) is then given by
![](download/temp/LaTeX3943924067815042491.png)
In this representation the matrix representations of all couplings are real including that of which is diagonal.
+For example, the and
operators couple the
and
states of AlO via the following matrix elements in the MOLPRO representation:
![](download/temp/LaTeX3270166283568502698.png)
where is a real number.
For we obtain
![](download/temp/LaTeX4898811939256642006.png)
Similarly, for the diagonal SO component coupling two components of the of AlO state, MOLPRO gives the following non-vanishing matrix elements:
![](download/temp/LaTeX9016390124907783958.png)
where is a real number. These couplings in the duo representation become real:
![](download/temp/LaTeX6334311070968018772.png)
The (non-vanishing) non-diagonal SO-matrix elements in the MOLPRO representation
+![](download/temp/LaTeX8962302739880374525.png)
( is a real number) in our representations (only non-vanishing and symmetrically unique) read
![](download/temp/LaTeX3983987705232049190.png)
The same relations can be obtained for the $B-A$ pair.
+ +The term of the Hamiltonian matrix depends only on the internuclear distance and thus can be included into the functional forms of the potential energy functions (diagonal
and
) or of the SO coupling terms (non-diagonal
) and thus can be excluded from our analysis.
The vibrational basis functions are prepared as solutions of the pure vibrational uncoupled eigen-problems for a given adiabatic electronic 'state'
![](download/temp/LaTeX2804480768552818038.png)
using the DVR method in the equidistant grid representation. Here is the corresponding potential energy function. For each electronic state once elects vmax lowest eigenfunctions as contracted basis functions for the rovibronic problem.
+The vibrational matrix elements for all -dependent terms appearing in the rovibronic Hamiltonian are evaluated numerically using the Simpson rule.
Finally, the spin-rovibronic functions are symmetrized to be eigenfunctions of the parity operator .
+A Hamiltonian matrix is generated and then diagonalized using the LAPACK routines DSYEV or DSYEVR as provided by the intel MKL libraries.
1 R. J. Le Roy, LEVEL 8.0 A Computer Program for Solving the Radial Schrodinger Equation
+for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report
+CP-663, http://leroy.uwaterloo.ca/programs/ (2007).
+2 H. Kato, Bull. Chem. Soc. Japan 66, 3203 (1993).