E. Norby Svendsen University of Wense 5000 Wense, Denmark
Energy Determination from the Electrostatic Theorem
Oneof the hig problems in teaching quantum chemistry is the fact that the calculations. even for small molecules, hecome too complicated. This is d u i to the difficult integrals, especiallv those arising from the electronic interaction operator llr,;. In this articie we avoid integrals of that type by calculating the molecular energy from the electrostatic theorem. This theorem is derived from the general Helhnann-Feynman theorem, which is a subject in several books on quantum chemistry (1-2). Here, and also in a recent paper on HOMO theory (3),the electrostatic theorem is mainly used to explain chemical bonding, whereas we merely use it as a tool for calculation. For a diatomic molecule the energy is a function of the internuclear distance R E(R) = w I A ( ~ , q ) l $= ) (+lp(R,q) + v(R,q)l$)
+
(1)
where is the molecular wavefunction and q denotes the electronic coordinates. The Hamilton operator is separated into its kinetic and potential energy parts. The derivative of the energy with respect to R is easily found
$=(*I($, 1+)
+2
From this equation i t follows that the binding energy can be determined directly from a knowledge of the force curve. This mav be comnared with the conventional method. where the ~, binding e n e r b is determined as a difference hetween the tothl enereies El..) . . and EtR. . ".I.. those numbers h ~ i -n rcalculated from the variational principle. As ~ o i n t e dout bv Hurlev (6).there is no a priori reason to prefe; the onemethod foriheoiher; only agreement with ex~erimentcan decide which is the best. The advantage of using kqn. (11) is that the calculations of F ( R ) only require calculation of intearals over one-electron operators, and also that the binding energy is determined by direct integration, and not as a small difference between two large total energies as is done in the conventional method. The main reason why the force method is not more often applied is due to the errors introduced in eqn. (3); the errors on F ( R ) are rather large. However, the form of the force curve is in good agreement with the exact force curve, so that force constants, given by dFldR, are found to agree within 20% with experiments (7). Let us illustrate the above with a calculation on the ground state of the hydrogen molecule. For ij. we take the simple LCAO wavefunction
-
(($)p-EI*) (2)
where q' are the fixed electron coordinates. For the exact wavefunction, satisfying the Schrodinger equation, we have
and eqn. ( 2 ) reduces to
which is the general form of the Hehann-Feynman theorem (4-5). If we measure q' along fixed axes, the kinetic operator is independent of R and eqn. ( 4 ) becomes
With this wavefunction the expression for the force curve becomes
(14) -..
--
The operator for the potential energy is given by
. . ".
".
.~,.,
where Z. and Z b are the charges on nuclei a and b, and N the number of electrons. Letting center a serve as the origin of the with. respect to R coordinate system, differentiation of gives Z, cos8,i Z.Zb
v
g=( * l ~ ~ l * -F )
(7)
This equation is usually named the electrostatic theorem. In the Born-Oppenheimer approximation the force on nucleus a is just the negative of the derivative of the energy as given in eqn. (7) d~ F ( R ) = - -dR =-
($lxl$)
aV
The integrals here are all one-electron integrals. The actual calculations are not tw difficult, and the details are given in the appendix. If we calculate F ( R ) for different values of R , we can draw a curve as shown in the figure. The experimental curve on the figure is obtained from some very accurate calculations of Kolos et al. ( 8 ) .From the figure we see that the form of the curve is good, and also that the binding energy, obtained from
1
D , = j Ro - d ~ = -I -RoF ( R ) ~ R
4
5
1.
(9)
6
7
8FXa.u.)
(10) (I1)
experiment
2. MO(i=l) 3. uo(r = 1.344)
which leads us to an expression for the binding energy a t the equilibrium distance Ro Ebinding= DP= E- - E(Rd
3
(8)
From this equation we have dE = -F(R)dR
2
F(R).IO~ Electrostatic force curves for lhe ground state of HI.
VOlUme 54. Number 6,June 1977 1 355
Scalingparameter. Internuclear Distance and Binding Energy for Conventional and Electrortatie Method
E
Method
R
DO
~ l u nl i t s i n a". a n k e n from e x p e r i m e n t %
ingly good compared with experiment and with the conventional method. I t is also interesting to note that this method does not give an upper bound for the energy, as it is the case for the variational method. Also i t must be noted that in the variational method the number of parameters which can be fixed is in principle unlimited, whereas in the force method one can only determine as many parameters as there are non-equivalent nuclei in the molecule. Appendix
Although the integrals in eqn. (14)cannot be done in one line, they can all be performed with a knowledge of elementary calculus. The first step is standard for two-center integrals, namely transformation integration of the force curve, given in the table, compares favorably to the conventional method. Usuallv the wavefunction is im~rovedbv introduction of some parameters ai.In the conventional method the parameters are determined by the requirements of the variational principle aElaai = 0. In the H2 calculation we often introduce a scaling ~arameter.so that the 1s orbital has the form
to elliptical coordinates. The integrals can then be evaluated by integrations by parts and they can all be expressed in terms of the standard functions (17)
Ei*(r)-=L ' q d t
We give below the final formulas Using the variation principle, this leads to an improvement of both the internuclear distance and the bindine.enerw .. as shown in the table. An alternative method to determine the parameter I would be to require the following equation to be fulfilled F(R)= -
($lzIt) av
=O
(16)
for R = Rn. In words, this means that the force on the nuclei must be zero a t equilibrium distance. This condition is of course satisfied for the exact wavefunction. For the actual case we find a {value of 1.344. The form of the force curve is shown in the figure. We see that the requirement of eqn. (16) brings the force curve verv close to the experimental one. The result of an integration gives avalue of D, of 0.178 au. This is surpris-
356 1 Journal of Chemical EducaHon
( lBb 1 9 11 % ) = $ (2 - ( 2 + 4j-R + 4 P R 2 ) ~ 2 f R ] (20)
(Is.
1 9 1 4 IS*) =
[(3 - 3t
+ t2)efEi*(2t)
+ ( y + log Zt)e-'(3 + 3t + t 2 ) - 3t(2 + t)e'']
(21)
where t = {R and y is Euler's constant Llteralure CPed ,'lfi,,. Pilar. Frank I..."RlementaryQusntumChemistry."MeGraw-Hill. 196R.$mre17P. I k b . H. M...l.CHRM. RDUC.. 52.:314 119751. Hellmnnn H.."Ein/~hruncindie Q u o n l ~ n r h r m i e , " S p r i n y e r .1531,pa~e2:W. Feynmnn. H. P.,Pl?s8.Rci,.. ifi.140119191. Hur1ey.A. C . . P ~ LHIIS . Sor. ILrndnn!. AWL. 170(18',41. I71 HenxUm M. L..and Kirtmnn, R..J. Chsm Phys., 44,119 119fifil. I81 Kc~lm.W a n d Wulniewicr, I...J Chem Phjii.. 41. :36fi:l 119641. In1 (:I1 141 151 161
