Energy Transfer between Molecules and Electronically Excited AtomsM. M. Shahin and S. R. Lipsky Citation: The Journal of Chemical Physics 41, 2021 (1964); doi: 10.1063/1.1726199 View online: http://dx.doi.org/10.1063/1.1726199 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/41/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Energy transfer between electronically excited zinc and calcium atoms J. Chem. Phys. 74, 1044 (1981); 10.1063/1.441237 Monopole effects on electronic excitation interactions between large molecules. I. Application to energytransfer in chlorophylls J. Chem. Phys. 67, 3901 (1977); 10.1063/1.435427 Energy transfer in collisions of excited Na atoms with NO molecules J. Chem. Phys. 67, 839 (1977); 10.1063/1.434848 Electronic Excitation Energy Transfer between Complex Ions J. Chem. Phys. 52, 4904 (1970); 10.1063/1.1673730 Transfer of Electronic Energy between a Metastable Argon Atom and a Nitrogen Molecule J. Chem. Phys. 47, 58 (1967); 10.1063/1.1711891

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 41. NUMBER 7 1 OCTOBER 1964

Energy Transfer between Molecules and Electronically Excited Atoms* M. M. SHAHINt AND S. R. LIPSKY

Department of Internal Medicine, Yale University School of Medicine, New Haven, Connecticut

(Received 1 June 1964)

The technique of competitive quenching has been used to determine the relative rate constants of O2,

N2, H2, CO2, and CH, for quenching of excited argon atoms formed both by the action of f3 radiation from a tritium source and also by low-energy electron impact. Calculations based on the mean lifetime of these species in the system show that with the exception of nitrogen the quenching process for all the other molecules is highly efficient and energy transfer occurs on every collision. Evidence is also presented to suggest that the excited species formed by collision with f3 rays are primarily in the IFI state and not the a PO,! metastable states.

INTRODUCTION

FORMATION of excited species of rare-gas atoms during the passage of ionizing radiation through

these gases has been known for some time. Energy transfer from these species to other atoms and molecules leading to the subsequent ionization of the latter has been studied by Jesse and Sadauskis (Jesse's effect) and further discussed by Platzman,l,2 The nature of these species has, however, not been established and hitherto remains largely a matter of speculation. Recently in an investigation which was primarily directed toward the study of Jesse's effect for ethylene in argon under varying electrostatic fields, it was noted that the presence of small quantities of oxygen in the system produced varying effects at different accelerat­ing potentials. Analysis of the experiments proved that it was feasible to measure the relative rate constant (or cross section) for energy transfer between the excited species of argon and molecules (e.g., H2, O2,

CO2, etc.) which do not exhibit the normal Jesse's effect owing to their high ionization potential (I.P.), but instead undergo chemical decomposition after the exchange of energy. A comparative study was there­fore undertaken in order to measure these rate con­stants and to observe whether any substantial differ­ences existed among the abilities of these molecules to quench such highly energetic species. Such a study is of interest in rare-gas sensitized radiolysis where energy transfer from the excited species of the rare gas would play an important part. Furthermore, oxygen being in a triplet ground state provided an excellent opportunity to examine the nature of these excited species, since it would be expected to have a substantially higher quenching cross section than that of, say either H2 or CO2, if the excited argon atoms were also in the triplet3,4 state.

* This work was supported by grants from the National Aero-nautics and Space Administration (NsG 192-61).

t Present address: Xerox Corporation, Webster, New York. 1 W. P. Jesse and J. Sadauskis, Phys. Rev. 100, 1755 (1955). 2 R. L. Platzman, J. Phys. Radium 21, 853 (1960). 3 The terms singlet and triplet are used here merely to dis­

tinguish the two different states of the excited argon atom. The weak L-S coupling for argon might possibly exclude the usual meaning of these terms in this case.

'G. Porter and M. R. Wright, Discussions Faraday Soc. 27, 18 (1959).

KINETIC ANALYSIS

When a gas mixture composed primarily of argon (>99%) and to a minor extent of two other polyatomic components Ml and M2 with ionization potentials 1ml and 1m2, is subjected to {3 radiation from a tritium source, it is expected that, assuming 1m2> Ea> 1ml, where Ea is the energy of the excited argon atom, the following reaction scheme would effectively describe the energy transfer events which lead to ionization in the systeml,2,5:

Ar~Ar+ and subsequently Ar2+ (1)

Ar~Ar* k2, (2)

Ar*~Ar (all self-quenching reactions) kd, (3)

Ar*+ Ml~Ar+ Ml *~Ml++e ki' (4) ~d .. ecomposltlOn k5, (5)

Ar*+M2~Ar+M2* k6, (6)

where Ar* represents the excited argon atom and the various values of k are the rate constants of the respec­tive reactions. kd describe the various steps through which the excited argon species will be destroyed in pure argon and involves terms which are dependent on argon-atom concentration. It will undoubtedly include steps such as any wall collisions, molecular metastable formation,! direct emission to the ground state for the optically allowed states, and excitation of any meta­stable states to the nearest radiative level through two­body collisions with argon atoms.5.6 The reaction of Ar**+Ar--+Ar2++e, where Ar** represents a higher energetic state than Ar*, is not considered here, since the large cross section measured for such a reaction' and the relatively high concentration of argon present will not permit Ar** species to establish significant competition for energy transfer with molecules such as Ml and M2 in the system. The presentation of Reactions (4) and (5), as both arise initially from the same excited Ml*, has been previously discussed7

and is also covered elsewhere.

6 L. B. Loeb, Basic Processes of Gaseous Electronics (University of California Press, Berkeley and Los Angeles, 1955).

6 A. H. Futch and F. A. Grant, Phys. Rev. 104, 356 (1956). 7 W. P. Jesse and R. L. Platzman, Nature 195, 790 (1962).

2021

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2022 M. M. SHAHIN AND S. R. LIPSKY

Application of the steady state kinetics to the excited species present in such a system under satura­tion current measurements, when either Ml alone or both Ml and M2 are added to pure argon, yields the following Stern & Volmer type equations, i.e.:

(a) when Ml alone is present,

(LlR1) = K2kiCml/[kd+ (k i+k6) Cm1], (I)

(b) when Ml and M2 are both present,

(LlR) 2= K2kiCml/[kd+k6Cm2 (ki+k6) Cm1], (II)

where (LlRh represents the increase in the rate of formation of ions when Ml type molecules are added to a system of pure argon, while (LlR)2 is the same differ­ence when argon initially contains a small fraction ( < 1 %) of M2 type molecules. The concentration terms Cml and Cm2 are given as (M 1) / (A) and (M2)/(A), respectively, and K2 is equal to k2(A).

If in the case where Eq. (I) is valid, Cm1 is varied and the value of 1/ (LlR)l is plotted vs l/Cml, a straight line would be obtainedl from which the ratio of intercept to slope yields (k i+k6)/kd. For the case where both Ml and M2 species are present, if Cm2 is kept constant and Cml varied, again a straight line is obtained for which the ratio of intercept to slope yields

(ki+k6)/kd 1 + (k6/kd) Cm2 .

From the knowledge of these two ratios and that of Cm2, the value of k6/kd can therefore be obtained. Thus by choosing the compound M2 at a fixed concen­tration Cm2, and measuring the difference in the level of ionization produced by the action of (3 radiation in two sets of experiments, namely in the presence and absence of M1, one could determine the ratio of its rate constant for energy transfer from excited species of argon to that which pertains to pure argon. The reference compound Ml chosen in these experiments is ethylene which has an ionization potential of 10.5 eV, well below the energy of the first excited state of argon.

EXPERIMENTAL

The microionization chamber used in this work has been previously described in detail elsewhere.8 The chamber is of coaxial design with 1 mm electrode spacing and a total volume of 280 JLliters. It uses a 200 mCi tritiated titanium foil as the source of radiation. The auxillary apparatus employed for this work was a conventional gas chromatographic unit (Model 10, Barber-Colman Company) equipped with a high­sensitivity electrometer (Model E-302, Gyra Com­pany, Chicago, Illinois) which was operated at ranges of 10-9 to 10-11 A. A Keithley Model 240 variable high-voltage supply was used as the source of power.

8 M. M. Shahin and S. R. Lipsky, Anal. Chern. 35, 467 (1963).

The ionization chamber was kept in a thermos tatted oven and maintained at 150±2°C. High-purity argon or argon containing controlled amounts of other gases was continuously passed through the chamber at a constant rate of 25 mljmin, and the flow was regulated by means of both an ordinary pressure regulator and an 8 ft coil of tin. i.d. glass tubing packed with Linde Molecular Sieve 5A, 60-80 mesh main­tained at 100°C. A secondary flow containing the diluted ethylene gas (2.71% of ethylene in argon) was allowed to mix with the main flow of argon just before entering the ionization chamber. This secondary flow was adjusted to the required concentration by the use of a pressure regulator and a 100 ft capillary coil of 0.010 in. i.d. steel tubing. Cylinders of various gas mixtures were made by Matheson Co. (East Ruther­ford, New Jersey) and subsequently analyzed by mass spectrometry. Both the main flow of argon and that of the secondary stream were measured by means of a calibrated flow meter.

SECONDARY EFFECTS INFLUENCING THE EXPERIMENTAL RESULTS

Previous experience with argon gas in such a system had shown that two other phenomena may affect the measurements of the (LlRh and (LlRh and can considerably change the interpretation of the data. These are: (1) effect of change in the drift velocity of the electrons as a result of the introduction of a polyatomic molecule into a system containing pure argon, and (2) the determination of the voltage re­quired to obtain saturation current, without sufficiently accelerating the secondary electrons in the system to cause further excitation of argon atoms on collision. These newly produced species can not be expected to be necessarily of the same type as those which originated from the action of (3 radiation. These two effects are further discussed below.

I. Effect of Electron Drift Velocity

In previous applications7 of ionization chambers to (3 radiolysis of highly purified argon gas, it had been noted that a saturation current plateau was never satis­factorily reached but instead the current was steadily increased with increasing accelerating potential. One major cause of this effect was considered to be the low drift velocity of the electrons in the system resulting from the relatively high energy which they acquire under the accelerating field (65% above 3 eV at E/p=l, Ref. 5, Chap. 4). Such high electron energies are normally encountered in systems of pure argon, krypton, or xenon since the probability of inelastic collisions between these (1-5 e V) electrons and the parent gas is relatively low. Introduction of trace amounts of polyatomic molecules to such a system, however, will drastically reduce the electron tempera­ture by allowing energy transfer to the low-lying rota-

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ENE R G Y T RAN S FER BET WEE N MOL E C U L E SAN D A TOM S 2023

tional and vibrational states of these molecules.9

This will consequently enhance the electron drift velocity and effectively increase the collected current, while the accelerating potential on the electrodes has remained unchanged.

This phenomenon was originally reported by Rossi and Staub9 and appears also in the investigations of Melton, Hurst, and BortnerlO where measurements of the value of W, the energy required for the formation of an ion pair, in the radiolysis of mixtures of argon with either oxygen, nitrogen, methane or carbon dioxide were made.

Since the present measurements require the deter­mination of the value of (k i+k6) jkd for the compound Mr, which in this case was chosen to be ethylene, the effect of the presence of this compound on the electron drift velocity and hence, its contribution to the total measured value of (t:J.Rh, had to be evaluated. Assum­ing that the effect is approximately the same for all polyatomic compounds,!l a calibration curve was therefore set up for the effect of various quantities (pp 104) of methane on the collected current at the particular potential used for the saturation current. This correction was applied to all measurements of (.:lR)r.

II. Saturation Current Voltage

As indicated above, the use of high accelerating potentials for obtaining saturation current in ioniza­tion chambers where argon, krypton or xenon are used as the main component gas, produces energetic elec­trons which in turn excite these atoms to their lowest excited states. The most important of these excited species are expected to be those of triplet metastable states (3 P2 and 3 Po) which are relatively long lived. The concentration of these species has been shown12 to increase considerably with the applied voltage such that their effect could subsequently overshadow that of the excited species which were originally formed by the action of {3 radiation. In order to ascertain the

TABLE I. Relative rate-constants for energy transfer from excited argon atom to various molecules; the value of (ki+k.) /ka for ethylene required for these determinations is taken as 2.0X 103•

Measurements are made at 300 V /ern.

Molecule 0,

1.2 2.3 0.1 0.95 1.3

9 B. B. Rossi and H. H. Staub, Ionization Cha~bers and Coun­ters; Experimental Techniques (McGraw-Hill Book Company, Inc., New York, 1949), Chap. 1.

10 C. E. Melton, G. S. Hurst, and T. E. Bortner, Phys. Rev. 96, 643 (1954).

11 Experiments with several polyatomic compounds showed that except for nitrogen, the observed effect is generally constant.

12 M. M. Shahin and S. R. Lipsky, Anal. Chern. 35, 1562 (1963) .

II-

10-

Q-

8-

7-

T < ~

6-

CI> .2

i~ 5--<3

4-

3-

2-

I

.oov,,;;'" I i ;.~/~

~~~ ;: / Y ,

, ,/ / ,

I 2

i'XI03

I :I

FIG. 1. Variation of the inverse of the current increment vs the inverse of the ethylene concentration for different saturation current voltages at constant oxygen concentration of 0.32 mole %.

applied voltage above which such an effect is no longer negligible, the Jesse's effect was studied for this system at varying applied potentials and has been reported elsewhere.12 Under the experimental conditions used here it was found that at 350 V jcm an already detectable number of these new species become avail­able, and at approximately 950 V jcm their effect completely overshadows that of the original excited species. At higher accelerating potentials, apparently a fraction of electrons in the system having energies considerably greater than the first excited state of argon will also become available, and thus probably produce other excited species. Accordingly, the ac­celerating potential used for the experiments reported here was chosen (300 V jcm) so as to avoid formation of the new species.

RESULTS AND DISCUSSION

Results of the measurements made with a number of gases together with other experimental parameters are given in Table I. The data for oxygen and hydrogen are shown on Figs. 1 and 2. The overall accuracy of the values of k6jkd is not considered to be better than 30% owing to the errors involved in the measurements of intercepts which are obtained through extrapolation of the best lines drawn through the data. These results were obtained at the saturation current voltage of 300 V jcm, where according to previous experiments12

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2024 M. M. SHAHIN AND S. R. LIPSKY

r----------------,-2.5 9-

8-

3 7-...... ::-0 0 6-a> .. 0 0

'" a 5-Ii? OJ

~ .l!i 4-

'< ;!l 3-

'" '2 )C

-14 2-

./1500V/CM

// //ICM / /K/OOV/CM

/0 /:/" / "/ / /

/>" ./ /

,,/ /'M. ~

I 2

I I :5 4

~xl03 C

I 5

I 6

-2.0

2 v "­::-

-1.5 ~

-/.O ~

)C

-I~ -0.5

FIG. 2. Variation of the inverse of the cur:ent increment. vs the inverse of the ethylene concentration for dIfreren~ saturation current voltages at constant hydrogen concentration of 0.27 mole %.

the field was not sufficiently high to cause the produc­tion of any detectable quantities of excited argon atoms through collisions of secondary electrons which have gained their energy from the field. The measurements thus refer to those excited atoms which are formed through the action of {3 radiation in the system. It can be observed that the values of k6/kd for the various molecules are of the same order of magnitude, except for nitrogen which appears to have a lower rate-con­stant ratio. This is in accord with other measurements of energy transfer which have been made for the efficiency of nitrogen to quench excited states of atoms and molecules.13 •14 The striking similarity between the values for H2, O2, CH4, and CO2, however suggests that the mechanism of quen~hing ma~ be :rery similar and is not affected by the dIfference In eIther the hydrogen content or the mUltiplicity of the quench­ing molecule. Oxygen being in a trip~et ~round ~tate is known4.15.16 to greatly favor de-exCItatIOn of trIplet atoms and molecules, and has rate-constants which are greater by 2-3 orders of magnitude than that of other molecules which contain paired electrons. Such high rate-constants apparently arise from the long-range interactions of two molecules (or atoms) with multi-

13 A. C. G. Mitchell and M. W. Zeman sky, Resonance Radiation and Excited Atoms (Cambridge University Press, New York, 1961), Chap. IV.

14 A. F. Trotman-Dickenson, Gas Kinetics (Butterworths Scientific Publications Ltd., London, 1955), Chap. 11.

Ii H. J. Groh, J. Chem. Phys. 21, 674 (1953). 18 G. J. Hoijtink, Mol. Phys. 3, 67 (1960).

plicities greater than one and may involve short-lived complexes which result in the breakdown of the selec­tion rule for the triplet-singlet forbidden transition. Such quenching processes often do not involve actual energy transfer to the quenching molecule. Interactions of singlet-singlet or singlet-triplet species, on the other hand, are negligible at long distances and the transfer of energy is not expected to occur until the molecules are in close proximity of each other, where the dipole­induced interactions become significant.17 Such a behavior has been known to occur for the excited or­ganic triplet molecules15 and is probably the explana­tion for the high quenching cross section of oxygen and nitric oxide (doublet molecule) for the excited triplet atom of mercury Hg (3Pl)' Such an interpretation of the data would indicate that the excited argon atoms formed by the action of {3 rays must be predominantly in a singlet state.

EXCITED TRIPLET ARGON ATOMS

To provide further evidence for such an assignment to these species, it would be appropriate to produce high concentrations of triplet argon atoms through electron bombardment within the system and then to further measure the rate-constant ratios for deactiva­tion of these species by oxygen and other molecules. The use of electron swarm technique18 to produce triplet excited atoms, though satisfactory at low gas pressures, is not quite successful in the present system owing to the phenomenon of radiation trapping (see below) and the inherently higher cross section for the formation excited singlet statesl9 as compared to the triplet states, for electrons of the same energy. High pressures used in this system will cause imprisonment of the resonance radiation and effectively lengthen the mean lifetime of the allowed singlet transitions, while the enhanced two-body and three-body deactivation processes will reduce the mean lifetime of the triplet excited species. Thus, the acceleration of electrons in such a system and their subsequent collision with the gas atoms is expected to produce a steady concentra­tion of the excited species composed primarily of the singlet state and to a lower extent of the triplet states. The latter is further expected to appear only during the interval of energies of the electron swarm which would cover the region just above the threshold for triplet formation, where the cross section for the formation of these species is relatively significant. Low proportions (a few percent) of the triplet species, produced by such a technique however, would still be expected to be detectable by the methods of oxygen deactivation, since the cross section for such a process

17 T. Forster, Discussions Faraday Soc. 27, 7 (1959). 18 Reference 4, Chap. VIII. 19 H. S. W. Massey and E. H. S. Burhop, Electronic and Atomic

Impact Phenomena (Oxford University Press, London, 1956), Chap. III.

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TABLE II. Values of k./kd for various molecules at different saturation current voltages. Measurements are based on (ki+k6)/kd =2.0X1Q3 for ethylene .

.. 10-3 X k./kd Applied field

in V/cm. 300 400 500 600 900 1100 1200 1500 2000

Hz 1.20 1.16 1.16 1.21 0.25

02 2.30 2.94 3.50 10.4 very large 2.0 1.90 0.90 0.70

CO2 0.95 0.92 1.10 0.96 0.67

is 2-3 orders of magnitude greater than the corre­sponding process for the singlet species.16 On this account, experiments similar to those of the previous section were carried out at varying accelerating fields above that required to reach saturation current. Previous experimentsl2 had shown that at sufficiently high voltage (about 800 V /cm), the concentration of these newly formed excited species will far exceed those initially produced by the action of (3 rays, and thus, they alone will effectively determine the measured rate constants. The results of these experiments are shown in Table II for oxygen, hydrogen, and carbon dioxide, and are also represented for the first two gases in Figs. 1 and 2.

It is observed that as the accelerating field is in­creased, the value of k6/kd for carbon dioxide or hy­drogen remains constant within the experimental error, while for oxygen it steadily rises until at 900 V / cm, it will no longer be possible to be measured by this technique and is thus recorded as infinity. Higher applied fields reduce the value of k6/kd for oxygen to its original value.

The data in Table II clearly show that a change specific to oxygen has taken place at about 900 V /cm. This change, reflected in the increase in the ratio ka/ kd for oxygen, can be ascribed as due to the higher deactivation cross section of the latter for the triplet argon atoms which may now be present in the system in sufficient number. However, such a conclusion assumes that the value of ka for the new species (triplet) has effectively remained unchanged as compared to that which pertains to the previous singlet species. A close examination of this term for these two species does in fact support such a view. Taking the known values20

of the two-body and three-body deactivation and diffusion processes which apply to the ap2, the lowest triplet excited argon atom, it can be calculated that at the pressure (1 atm) used in these experiments the mean lifetime T = 1/ kd of these species is as low as 2.0X 10-7 sec. On the other hand the natural lifetime of the excited singlet atoms I PI, expected to be the rate determining step for the self-destruction of these species, i.e. Tns.t= l/kd, has been calculated21 to be

20A. V. Phelps and J. P. Molnar, Phys. Rev. 89,1202 (1953). 21 R. Knox, Phys. Rev. 110, 375 (1958).

3.0X 10-9 secs. However, because of the high pressure used in this system as mentioned above, radiation trapping is expected to effectively increase the mean lifetime of these species and make the latter closer to that for the triplet metastable species. Using Hol­stein's theory4 to calculate the effect of this radiation imprisonment an effective lifetime of lOX 10-7 secs can be obtained for the I PI singlet species. Considering the limits of both the experimental and theoretical calculations, these results are in excellent agreement with each other. It therefore becomes apparent that the variation in the value of ka/kd observed for oxygen does in fact reflect higher deactivation cross section of the latter as compared to those of hydrogen and carbon dioxide for the triplet excited states of argon. Further, it may be concluded that the deactivation cross section of molecules such as hydrogen or carbon dioxide does not significantly differ for the singlet or the triplet species.

Table II, also shows that beyond 900 V / cm, the quenching rate constants begin to diminish slowly as the applied field is increased. This may be due to the importance of secondary processes such as direct ionization of the quenching molecules by energetic electrons in the system. This phenomenon is common to all the quenching gases which have been used.

THEORETICAL CONSIDERATION AND ASSIGNMENT OF THE EXCITED STATE

In a system where keY {3 rays are used as a source of radiation, it can be assumed that by far the greater majority of excitation and ionization acts occur through impacts of high energy electrons22 (e.g. E»ionization potential of the atom). Under such conditions, the use of the Born-Oppenheimer approxi­mation in the theory of inelastic collisions permits certain general conclusions which appear in accord with the findings of these experiments. For example, at electron energies very much greater than the threshold for excitation, all inelastic cross sections appear to fall off with increase in electron energy, although the rate of fall is more rapid for transitions involving change of

22 Calculations for the case of helium [R. L. Platzman, Intern. J. Appl. Radiation and Isotopes 10, 116 (1961) ] show that for 10 KeV electrons, 82% of the ionization is by electrons of more than 10 eV in energy.

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2026 M. M. SHAHIN AND S. R. LIPSKY

multiplicity (e.g. singlet-triplet) than for those which mayor may not be optically allowed but require no change in the multiplicity of the atom. Consequently, where high-energy electrons are involved in the excita­tion of an atom, transitions to triplet states are less likely than those to singlet states. As a result the actual magnitude of the excitation cross section appears greater for the singlet than the triplet transitions at these electron energies. These conclusions are borne out by the agreement between theoretical and experi­mental cross sections which have been given for the helium atom.19 Thus it may be concluded that the formation of singlet excited states of argon by the action of {3 rays is not surprising and is indeed favored. Specifically, however, to assign any particular state to these species would require some knowledge of their structure although the lowest excited configuration is the most likely candidate.

THE JOURNAL OF CHEMICAL PHYSICS

In experiments where energy transfer from these species could ionize a test molecule, an upper limit for the energy of this state could be found through stepwise increase in the ionization potential of the test molecule until the available energy is no longer sufficient to ionize it. Experiments both in these laboratories and elsewherelo place the limit for argon between 11.6 eV (I.P. of C2H6) and 11.8 eV (I.P. of CCI2F2), subject to the accuracy of the determination of these values. Such an estimate would preclude all excited configurations of argon atoms except the first, namely 3p54s which is comprised of four closely spaced levels, three of which are triplet (3 P2, 3 PI, and 3PO with energies of 11.54, 11.62, and 11.72 eV, respectively), and one singlet state (IPI with an energy of 11.82 eV). Thus if the foregoing interpretation of the results is correct the excited argon atoms are expected to be mainly in the I PI state.

VOLUME 41, NUMBER 7 1 OCTOBER 1964

Constant Energy and Minimum Energy Orthogonalization * HUBERT W. JOY

Metals and Ceramics Division, Oak Ridge Natiunal Laboratory, Oak Ridge, Tennessee

AND

L. J. SCHAAD t Department of Chemistry, Vanderbilt University, Nashville, Tennessee

AND

GEORGE S. HANDLERt, §

Department of Chemistry, Tufts University, Medford, Massachusetts

(Received 28 February 1964)

The problem of orthogonalizing a set of orbitals is considered, noting that the energy calculated for a determinantal wavefunction is often changed by such a procedure. Rules are given for finding whether a particular determinantal wavefunction will, in fact, yield different results on orthogonalization of its basis set. A new orthogonalization procedure, minimum energy orthogonalization, is illustrated, and some of its possible uses are examined.

INTRODUCTION

I N atomic and molecular calculations one often wishes to use a conceptually simple set of basis functions,

but these in general will not be orthogonal and will lead to complicated energy calculations. One might transform the original functions to a computationally simple orthogonal set; however, this process will often increase the computed energy. by mixing in high-energy components. I

* Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation.

t Summer research participant, Metals and Ceramics Division, Oak Ridge National Laboratory.

t Consultant, Metals and Ceramics Division, Oak Ridge National Laboratory.

§ Present address: Department of Chemistry, University of California (San Diego), La Jolla, California.

1 J. C. Slater, J. Chem. Phys. 19, 220 (1951).

The matrix C which transforms a set of nonorthog­onal orbitals {f;} to an orthonormal set {4>;} is not unique. C transforms the overlap matrix S

to a unit matrix, and any matrix CU, where U is unitary, has this property. At least three orthogonali­zation procedures are common: (a) Schmidt orthog­onalization,2 where C is triangular; (b) symmetric orthogonalization,3 where C is Hermitian, and (c) can­onical orthogonalization, where C is a particular modal

2 A. S. Householder, Principles of Numerical Analysis (McGraw­Hill Book Company, Inc., New York, 1953), p. 72.

3 P.-O. L5wdin, Advan. Phys. 5, 44 (1956).

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