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<!DOCTYPE html>
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<title>Sergey Yurchenko : Solving the rovibronic Schrödinger equation for the nuclear motion</title>
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Sergey Yurchenko : Solving the rovibronic Schrödinger equation for the nuclear motion
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Created by <span class='author'> Sergey Yurchenko</span>, last modified on Oct 31, 2013
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<p>Duo is a general program for solving a coupled rovibronic problem for diatomics.</p>
<p>Within the Born-Oppenheimer approximation the spin-orbit (SO) free<br />
Hamiltonian of the diatomic problem in the absence of the hyperfine splitting<br />
is given by <sup>2</sup>:</p>
<img src="download/temp/LaTeX2692729341814108318.png"/>
<p>where <img src="download/temp/LaTeX1374731485769934316.png"/> is the electronic part, <img src="download/temp/LaTeX2433279306261344106.png"/> is the rotational part and <img src="download/temp/LaTeX5287773473069865075.png"/> is the vibrational part.<br />
The rotational angular momentum <img src="download/temp/LaTeX8968004807919143273.png"/> is perpendicular to the internuclear axis. Considering Hund's case <em>a</em> and representing the rotational angular momentum as</p>
<img src="download/temp/LaTeX6769203689468309757.png"/>
<p>in terms of the total <img src="download/temp/LaTeX5988856084570285479.png"/>, electronic <img src="download/temp/LaTeX2922864761007324481.png"/>, and spin <img src="download/temp/LaTeX8482434831397942704.png"/> angular momenta and choosing the body-fixed <img src="download/temp/LaTeX4560767171869173798.png"/> axis along the internuclear axis the rotational part is given by</p>
<img src="download/temp/LaTeX4985196685252994968.png"/>
<p>where <img src="download/temp/LaTeX2125314434822097478.png"/>, <img src="download/temp/LaTeX4620271591331818195.png"/> and <img src="download/temp/LaTeX7001061076666749564.png"/> are the corresponding momentum ladder operators.</p>
<p>The duo basis set functions are chosen as the product</p>
<img src="download/temp/LaTeX8164832238879656507.png"/>
<p>where <img src="download/temp/LaTeX1189512006902003190.png"/> is the electronic function, <img src="download/temp/LaTeX4592312314073701046.png"/> is the rotational function, <img src="download/temp/LaTeX8675186813472710603.png"/> is the vibrational function; 'state' is the label identifying the electronic state; <img src="download/temp/LaTeX904113048198662933.png"/>, <img src="download/temp/LaTeX3839272838389558649.png"/>, and <img src="download/temp/LaTeX5156350684888801100.png"/> are the <img src="download/temp/LaTeX208507749196402145.png"/> axis projections of the electronic, spin and total angular momenta, respectively, and <img src="download/temp/LaTeX6501366754793362016.png"/>; <img src="download/temp/LaTeX5740908255765503230.png"/> is the projection of the total angular momentum along the laboratory axis <img src="download/temp/LaTeX3047735604315068366.png"/>; <img src="download/temp/LaTeX7915928944797145993.png"/> is the vibrational quantum number.</p>
<p>For the rotational basis set the rigid rotor functions are used, with the following matrix elements:</p>
<img src="download/temp/LaTeX2957261370005360606.png"/>
<p>where <img src="download/temp/LaTeX2953088953541628867.png"/> is omitted for the simplicity.</p>
<p>The electronic basis functions appear in the solution only implicitly as matrix elements of different components of the operator <img src="download/temp/LaTeX8521926696661495881.png"/> as well as of the spin-orbit contributions. We choose these basis functions to satisfy the following conditions^2^:</p>
<img src="download/temp/LaTeX5307144792322887259.png"/>
<p>where <img src="download/temp/LaTeX4976588540765127444.png"/> is a reflection through the molecular-fixed <img src="download/temp/LaTeX8304545636593666996.png"/>-plane (parity operator) and <img src="download/temp/LaTeX2854438776583206914.png"/> for <img src="download/temp/LaTeX6836014443712380309.png"/> states and <img src="download/temp/LaTeX4281687516611823155.png"/> for all other states. The following non-vanishing matrix elements of the spin operators are valid:</p>
<img src="download/temp/LaTeX6457220849845493965.png"/>
<p>where the label 'states' is omitted for the simplicity.</p>
<p>For example, the non-vanishing and symmetrically unique matrix elements of the ladder angular momentum operators <img src="download/temp/LaTeX4735759893083896316.png"/> as well as of the SO-matrix elements on this basis functions for the three lowest states of AlO <img src="download/temp/LaTeX1159418765929263877.png"/>, <img src="download/temp/LaTeX7069213538161000441.png"/> and <img src="download/temp/LaTeX1064988572674952451.png"/> are obtained using MOLPRO and can be summarized as follows:</p>
<img src="download/temp/LaTeX5836138745718601059.png"/>
<p>where <img src="download/temp/LaTeX8181176953970280968.png"/> is omitted for the simplicity.<br />
All other non-vanishing matrix elements can be obtained by applying the symmetric properties upon the reflection (see above).</p>
<p>It should be noted for <img src="download/temp/LaTeX6746474379497378469.png"/> the MOLPRO functions are not eigenfunctions of <img src="download/temp/LaTeX4502650293584219960.png"/> as required by the equation above. For example, the electronic state <img src="download/temp/LaTeX1447164918032902140.png"/> gives rise to two degenerate solutions labelled in MOLPRO as <img src="download/temp/LaTeX2266975164843202178.png"/> and <img src="download/temp/LaTeX4508541783344653704.png"/>. The corresponding matrix representation of the <img src="download/temp/LaTeX8949966229816269524.png"/> operator is not diagonal as computed by MOLPRO with the matrix elements is usually given by</p>
<img src="download/temp/LaTeX8778477990539368986.png"/>
<p>Besides the matrix elements of <img src="download/temp/LaTeX1916198807899282963.png"/> and <img src="download/temp/LaTeX8203546839009169993.png"/> as well as the spin-orbit matrix elements are complex numbers in the MOLPRO representation.<br />
The transformation from <img src="download/temp/LaTeX3116329121140092668.png"/> and <img src="download/temp/LaTeX4162665952592549342.png"/> (MOLPRO) to our representation <img src="download/temp/LaTeX4483087236980926278.png"/> (duo) is then given by</p>
<img src="download/temp/LaTeX3943924067815042491.png"/>
<p>In this representation the matrix representations of all couplings are real including that of <img src="download/temp/LaTeX8191389906650066627.png"/> which is diagonal.<br />
For example, the <img src="download/temp/LaTeX6181655575512959522.png"/> and <img src="download/temp/LaTeX5755403592565244100.png"/> operators couple the <img src="download/temp/LaTeX2213428100272282003.png"/> and <img src="download/temp/LaTeX7471506092673354338.png"/> states of AlO via the following matrix elements in the MOLPRO representation:</p>
<img src="download/temp/LaTeX3270166283568502698.png"/>
<p>where <img src="download/temp/LaTeX8398267673538105879.png"/> is a real number.</p>
<p>For <img src="download/temp/LaTeX7902452012369652673.png"/> we obtain</p>
<img src="download/temp/LaTeX4898811939256642006.png"/>
<p>Similarly, for the diagonal SO component coupling two components of the <img src="download/temp/LaTeX1680843422102632345.png"/> of AlO state, MOLPRO gives the following non-vanishing matrix elements:</p>
<img src="download/temp/LaTeX9016390124907783958.png"/>
<p>where <img src="download/temp/LaTeX401529467149484829.png"/> is a real number. These couplings in the duo representation become real:</p>
<img src="download/temp/LaTeX6334311070968018772.png"/>
<p>The (non-vanishing) non-diagonal SO-matrix elements in the MOLPRO representation</p>
<img src="download/temp/LaTeX8962302739880374525.png"/>
<p>(<img src="download/temp/LaTeX2429415034757130162.png"/> is a real number) in our representations (only non-vanishing and symmetrically unique) read</p>
<img src="download/temp/LaTeX3983987705232049190.png"/>
<p>The same relations can be obtained for the $B-A$ pair.</p>
<p>The <img src="download/temp/LaTeX3139721093327912400.png"/> 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 <img src="download/temp/LaTeX6235713817979751734.png"/> and <img src="download/temp/LaTeX173202487278737021.png"/>) or of the SO coupling terms (non-diagonal <img src="download/temp/LaTeX2062539812721079874.png"/>) and thus can be excluded from our analysis.</p>
<p>The vibrational basis functions <img src="download/temp/LaTeX1854690311273396413.png"/> are prepared as solutions of the pure vibrational uncoupled eigen-problems for a given adiabatic electronic 'state'</p>
<img src="download/temp/LaTeX2804480768552818038.png"/>
<p>using the DVR method in the equidistant grid representation. Here <img src="download/temp/LaTeX6042012672728598606.png"/> is the corresponding potential energy function. For each electronic state once elects vmax lowest eigenfunctions as contracted basis functions for the rovibronic problem.<br />
The vibrational matrix elements for all <img src="download/temp/LaTeX2501206966256259836.png"/>-dependent terms appearing in the rovibronic Hamiltonian are evaluated numerically using the Simpson rule.</p>
<p>Finally, the spin-rovibronic functions are symmetrized to be eigenfunctions of the parity operator <img src="download/temp/LaTeX3715235889939841815.png"/>.<br />
A Hamiltonian matrix is generated and then diagonalized using the LAPACK routines DSYEV or DSYEVR as provided by the intel MKL libraries.</p>
<p><sup>1</sup> R. J. Le Roy, LEVEL 8.0 A Computer Program for Solving the Radial Schrodinger Equation<br />
for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report<br />
CP-663, <a class="external-link" href="http:https://leroy.uwaterloo.ca/programs/" rel="">http:https://leroy.uwaterloo.ca/programs/</a> (2007).<br />
<sup>2</sup> H. Kato, Bull. Chem. Soc. Japan 66, 3203 (1993).</p>
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