Ab Initio Calculation of the E 1 g and a 3 g + States of the Hydrogen MoleculeJohn Gerhauser and Howard S. Taylor Citation: The Journal of Chemical Physics 42, 3621 (1965); doi: 10.1063/1.1695768 View online: http://dx.doi.org/10.1063/1.1695768 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/42/10?ver=pdfcov Published by the AIP Publishing

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 42, NUMBER 10 15 MAY 1965

Ab Initio Calculation of the E 12;g and a 32;g+ States of the Hydrogen Molecule

JOHN GERHAUSER

Rocketdyne, A Division of North American Amation, Inc., Canoga Park, California

AND

HOWARD S. TAYLOR

Department of Chemistry, University of Southern California, Los Angeles, California, and Jet Propulsion Laboratory, Pasadena, California

(Received 3 December 1964)

Molecular orbital calculations have been made of the potential function for the E 12;.+ and a 32;.+ excited state of the hydrogen molecule. Minimum energies of -0.71643 and -0.72730 a.u. were obtained, respec­tively. It was found that states separate into two hydrogen atoms-that is, one electron is associated with each nucleus. The potential curve for a 32:.+ is a normal stable species curve, but the E 1~.+ curve has an unusual hump in agreement with the results of Davidson.

THE method of Harris and Taylor1,2 has been used to calculate the wavefunctions and potential curve

for two excited states of H2; E 12;g+ and a 32;g+.3 The details of the calculation procedure are given in Refs. 1 and 2. Wavefunctions and theoretical potential curves are necessary in the calculation of the molecular parameters: electron distributions, spectral interpreta­tion, absolute intensities, electric moments, and electron correlation. Also, the wavefunction is a starting point for the time-dependent perturbation equation which can be used to calculate more complicated processes. The configuration-interaction wavefunctions can be compared to the James-and-Coolidge-type calculations, and the comparative value for excited states of the different calculation techniques determined.

The configuration interaction method, with variation of screening parameters in open shell configurations, converge very rapidly (four configurations give over 0.995 of the energy). The conceptual advantage may thus be the most significant feature of this approach.

The earliest calculation of excited states for hydrogen was the James and Coolidge4 calculation of 32;0+' Until recently, after Shull and Lowdin6 showed that the orthogonalization to the ground state is unnecessary if the correct root of the secular determinant is used, few excited-state calculations were made because of the difficulty of orthogonalizing the trial function with the true ground state. Since then, a number of calcu­lations have been made; the hydrogen molecule calcu­lations of the two states discussed here are Kolos and Roothaan6 for the state E 12;g+; Davidson7 for E 12;g+;

Zung and DuncanS for E 12;g+ and a 32;0+; Kato, Hayes,

1 F. E. Harris, J. Chern. Phys. 32, 3 (1960). 2 F. E. Harris and H. S. Taylor, J. Chern. Phys. 38, 2591 (1963). 3 These are the next-to-lowest and lowest states of their respec-

tive symmetries. 4 H. M. James and A. S. Coolidge, J. Chern. Phys. 6, 730

(1938). & H. Shull and P.-O. Uiwdin, Phys. Rev. 110, 1466 (1958). 6 W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys. 32, 219

(1960). 7 E. R. Davidson, J. Chern. Phys. 33, 1577 (1960); 35, 1189

(1961). 8 J. T. Zung and A. B. F. Duncan, J. Chern. Phys. 36, 2140

(1962) .

and Duncan9 for E 12;0+; Tamassey-LenteilO for a 32;g+;

and Huzinagall for a 32;g+. The calculations involving these states (E 12;g+ and a 32;g+) are summarized in Table I.

DISCUSSION OF RESULTS

The potential curve for the state E 12;g+ has an un­usual hump according to the calculation of Davidson.7

The hump appears to be caused by the interaction of a

-.60

-.62

-.64

';jj u ·f -.66

t:~ ~ -.68 2- Conllgurotion .. velunetlon

-.70 4- Configurotion wavefunction

-.72':---'_-"_-'-_-:'-_.1.---''----'-_-'--' ~ W ~. ~ M ~ ro ~ M

R(cw.)

FIG. 1. Theoretical potential curve E 1~. +.

number of excited states. Thus, it is of interest to see a second independent calculation of the state. The results are tabulated in Table II and plotted in Fig. 1. As can be seen, the form of the potential curve our work agrees with Davidson.

The state a 32;0+ has approximately the same energy as E 12;0+' It has a regular stable potential curve as given in Table III and plotted in Fig. 2.

Davidson gave an extensive discussion of the H2 12;g+

state and because of our agreement with the form of the potential curve he exhibits we restrict our discus-

g Y. Kato, E. F. Hayes, and A. B. F. Duncan, J. Chern. Phys. 41,986 (1964).

10 I. Tamassey-Lentei, Acta Phys. Acad. Sci. Hung. 12, 199 (1960) .

11 S. Huzinaga, Progr. Theoret. Phys. (Kyoto) 17, 162 (1957).

3621

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3622 J. GERHAUSER AND H. S. TAYLOR

TABLE I. E 12:.+ calculations.

R (a.u.) KR D ZD This work four conf. Two conf.

A. Energy of E 12:.+

0.5 1.0 1.25 1.5 1.8 1.85 1.9 1.95 2.0 2.25 2.50 2.75 3.0 3.25 3.30 3.5 3.75 4.0 4.25 4.30 4.5 4.75 5.0

1.0 1. 80 1. 87 1.90 2.00 3.00 4.00 5.00

10.00

-0.71500

-0.71635

-0.71612

0.1615 -0.57543

-0.69882 -0.71508

-0.71618&

-0.71587

-0.70339 -0.68683 -0.68443

-0.68797 -0.69447 -0.69825 -0.70049

-0.69914 -0.69685

& -0.7170 for 32-function minimization; Kato, Hays, and Duncan, -0.69688, Experimental -0.7181, Energy in hartrees, Huzinaga -0.69913, James and

sion to one short point. In our result only four config­urations were found that contributed significantly. Moreover, these four configurations were the same at all internuclear separation. In contrast Davidson found different types of functions at the various inflection points. There is of course a great change in the screening

.. .. ~ o £ IIJ

-~9·r-------------------------------~

-.61

10 aD 9.0 100

R (o.u.)

FIG. 2. Theoretical potential curve a 32).+.

-0.66574 -0.70320 -0.70216

-0.69174 -0.71545 -0.71459 -0.71578

-0.69211 -0.71643 -0.7158 -0.71626

-0.69125 -0.71570 -0.71428 -0.70902 -0.70163 -0.69366 -0.68722 -0.68295

-0.68709 -0.68333 -0.68724 -0.68848

-0.70522 -0.69621 -0.69474

-0.69959 -0.68992

-0.59197 -0.71372

-0.72730 -0.71329 -0.71169 -0.72602

-0.69372 -0.65758 -0.63330 -0.62255

Coolidge -0.7357, Tamassey-Lentei -0.6731, Experimental -0.7371. Energy in hartrees.

parameters of the functions and qualitatively these indicate that at the outer minimum the species is not very molecular; i.e., the electron seems to be localized, one per atom as opposed to Davidson's H+H- descrip­tion. As expected, extensive variation of nonlinear parameters gives somewhat better answers with far fewer configurations.

It is also interesting that at the inner minimum our wavefunction is superior to the 40-term Kolos and Roothaan function but not as good as Davidson's 32-term wavefunction. Clearly including r12 without split orbitals is not as important as extensive variation of nonlinear parameters in a very flexible function with split orbitals and without r12. The reason is clear; the one electron molecular orbital shapes are more im­portant than correlation in two electron excited states where one electron is "up" and the other "down." Parameter variation in a flexible molecular orbital seems to be better for orbital shaping than does the "rI2"

method with extensive configuration interaction. How­ever, as noted below, wavefunction convergence diffi­culties were important after the first four configurations. Davidson's convergence also was slow; he needed 32 configurations to get an energy of -0.71700 at R= 1.92,

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EI2:. + AND a 32: g + STATES OF H2 3623

TABLE II. 12:. States.

Configurations

R (a.u.) 1st 2nd

(:) (~~) /l t /l t

1.50 0.40/0.34 0.34/0.15 1.09/-0.72 1.09/-0.72

1. 80 0.74/1.08 0.36/0.15 1.22/-0.40 1.24/-0.80

0.73/0.90 0.36/0.10 1.24/-0.57 1.27/-0.81

1.85

1.90 0.72/1.11 0.37/0.07 1.27/-0.57 1.29/-0.86

1. 95 0.64/0.60 0.39/0.06 1.33/-0.99 1.32/-0.93

2.00 0.63/0.60 0.41/0.12 1.34/-1.05 1.35/-0.95

2.25 0.94/1.15 0.42/0.18 1.45/-0.71 1.46/-0.99

3.30 1.35/2.04 0.76/0.11 1. 70/-1. 62 1.93/-1.53

4.30 1. 25/1. 99 1.44/-0.11 2.31/-2.40 2.45/-2.21

5.00 1.08/2.48 1.66/-0.14 2.64/-2.60 2.73/-2.60

where -0.7181 a.u. is the experimentally observed value. Davidson and we agree that the apparent source of difficulty is the shape of the outer shell function which is a spherically symmetric function centered about the center of the molecule. This is very difficult to achieve with the elliptical bases of the type used in this and Davidson's paper. Kato, Hayes, and Duncan have tried such a three-centered function, i.e., one molecular orbital is two centered and the other one centered. They did not do an elaborate calculation but their result bears out the above conclusions.

In the calculation of the state a 3~g+, there are some interesting comparisons. The calculation here gives the best results for the potential curve for this state, but the calculation by James and Coolidge is still the most accurate calculation at the minimum. Despite this state being automatically orthogonal to the ground state, few calculations for it have been done. Except close to the minimum we have the only potential curve calculation for the state. The state is close to E l~g+ in energy, and the same trial wavefunctions are adequate to describe both states. However, two functions do not give nearly as good an energy in a 3~g+ as in E l~g+. Again the separation is two hydrogen atoms.

WAVEFUNCTION

The approximate wavefunctions for the excited states E l~Q+ and a 3~Q+ of the hydrogen molecule have been

3rd 4th

(~~) eo 1) 20-1

/l r /l t

0.93/0.24 3.70/0.88 1.14/-0.43 3.70/0.88 0.70320

0.93/0.23 3.49/0.88 0.85/-0.34 3.49/0.88 0.71545

0.84/0.75 3.57/0.84 1. 06/ -1. 53 3.57/0.84 0.71578

0.92/0.23 3.63/0.88 1.12/-0.39 3.63/0.88 0.71643

0.93/0.24 3.70/0.88 1.14/-0.43 3.70/0.88 0.71626

1. 00/0. 30 3.78/0.88 1.10/ -0.47 3.78/0.88 0.71570

1.19/0.06 4.23/0.78 1.07/-0.30 4.23/0.78 0.70981

1. 90/0. 42 2.25/1. 48 2.04/-1.61 2.26/1.48 0.68709

2.38/0.95 4.85/1.80 2.44/-2.60 4.85/1. 80 0.70522

1.16/0.59 5.20/2.00 2.12/-3.24 5.20/2.00 0.69959

developed by the method of Taylor and Harris.l.2 The projection of the primative function for the proper symmetry is done on a single configuration before minimization. In general the angular momentum and parity symmetry are included by choice of primative functions while g-u and antisymmetrization symmetry

TABLE III. Two configuration function for a 32:; ••

(~~) e~) R /l t /l r E

1.0 0.78/0.48 1.28/-0.44 0.16/-0.10 0.12/0.12 -0.59197

1.87 1.24/0.81 1.61/-0.85 0.25/-0.16 0.17/0.22 -0.72730

2.0 1.41/0.78 1. 98/-0. 93 1.11/0.11 0.31/0.13 -0.72602

3.0 1.40/0.52 2.50/-1.34 1.66/-1.0 0.46/0.32 -0.69372

4.0 1.64/0.46 3.02/-1.76 2.21/-1.47 0.62/0.52 -0.65758

5.0 2.50/1.05 3.36/-2.18 2.96/-2.30 0.62/1. 03 -0.63330

10.0 7.14/5.78 1.84/-1.64 5.22/-4.4 4.86/4.87 -0.62255

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3624 J. GERHAUSER AND H. S. TAYLOR

are done automatically by the computer program. The spatial one-electron function is an open-shell function defined in confocal elliptical coordinates (1:::;~:::; 00; -1:::;'11:::;1; 0:::;cp:::;211') with the state parameters n, m, and v; and the screening parameters 5 and 5:

where I~, 5; nmv)='S.(n)H(m)q,(v) ,

'S.(n) = exp{ - ~O~n,

H(m) = exp{ -51JI71m,

q,(v) = exp{ ivcp} [(~-1)(1-712) JI.1/2.

o is positive definite, nand m are nonnegative integers, and v is any integer. [The wavefunctional form is abbreviated

(n1m1v1) n2m2v2

for two electrons.JI2 The wavefunction is neither nor­malized nor orthogonalized. The orbital exponents 5 and 5 are minimized by a search procedure in which the energy calculation is done for varied parametric values. The configuration interaction variation is by the usual secular determinant.

The choice of trial wavefunctions is extremely im­portant for a good calculation. The first configurations to include are the state configurations and are obvious. It should be noted that if the function is developed from atomic orbitals, extraneous terms may be in­cluded. For example, the 2s state goes to (100) + (010). The (010) function is of negligible importance in both the l~g and 3~g states. Next shaping of the electron distribution and electron correlation needs to be in­cluded. The usual technique for this is inclusion of excited states developed from atomic functions, corre­lation functions (i.e., a nonzero v quantum number), or the inclusion of 1'12 terms. For the l~g+ state

(20 1) 20-1

is slightly better than

as the correlation function. It is expected that the 2p., 2p. function would also mix. However, 3p. 1s is more effective. The two functions affect the wavefunction in the same way; inclusion of both is no better than the 3p., 1s function. For example, using 2p., 2p. the energy at R= 1.9 is -0.7160 a.u. compared to 3p., 1s energy of -0.7164; at R=4.3, the energies are -0.6999 and -0.7052, respectively.

11 We indicate a wavefunction as (nmJl) for the values of the state parameters.

The coefficients of the configurations obtained from minimization do not quantitatively show the impor­tance of a configuration since the functions used are nonorthogonal. Qualitatively, for the l~g+ state, the configuration

is always the dominant configuration; the next most important configuration is

the configuration

is of minor importance for the first minimum, and of comparable importance to

at the second minimum;

(20 1) 20-1

is always of minor importance. In the 3~g+ state, the configuration

is always dominant. The vibration-rotation levels of both the states have

been considered, and are not of a trivial nature. Dieke13

and Herzberg14 in the 11;" case, and HerzbergI4 in the 3~g case, were unable to fit the experimental data to simple three-parameter Morse curves. In the l~g case it would be indeed surprising if simple fits were possible; the double minima forbid this. Davidson, whose potential curve is essentially the same as ours, has done a numeri­cal integration of the rotation-vibration wave equation for the given potential and found, as did Dieke with the experimental data, that w. depended strongly on vibrational level. For 3~g, Herzberg14 shows that w.x. and w.y. are not negligible. In agreement with this, we find that an attempt to fit a Morse curve to the poten­tial gives an ambiguous force constant of 2590 cm-1:::;

w.:::;2970 cm-I .

III G. H. Dieke, J. Mol. Spectry. 2, 494 (1958). I. G. Herzberg, Spectra of Diatomic Molecules (D. Van Nos­trand Company, Inc., New York, 1950).

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