Multiperturbation approach to potential energy surfaces for polyatomic molecules Donald H. Galvan, Moh’d AbuJafar, and Frank C. Sanders Citation: J. Chem. Phys. 102, 4919 (1995); doi: 10.1063/1.469540 View online: http://dx.doi.org/10.1063/1.469540 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v102/i12 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Multiperturbation approach to potential energy surfaces for polyatomic molecules Donald H. Galvan Instituto de Fisica de UNAM, Laboratorio de Ensenada, Apto. Postal 2681, Ensenada, Baja California 22800, Mexico Moh’d Abu-Jafara) and Frank C. Sanders Department of Physics and Molecular Science Program, Southern Illinois University at Carbondale, Carbondale, Illinois 62901-4401 ~Received 22 August 1994; accepted 12 December 1994! In Z-dependent perturbation theory, the lowest-order wave functions for a polyatomic molecule are not only independent of the nuclear charges, but also of the total number of nuclear centers and electrons in the molecule. The complexity of the problem is then determined by the highest order retained in the calculation. Choosing the simplest possible unperturbed Hamiltonian, we describe an n-electron, m-center polyatomic molecule as n ‘‘hydrogenic’’ electrons on a single center perturbed by electron–electron and electron–nucleus Coulomb interactions. With this H 0 , the first-order wave function for any polyatomic molecule will be a sum of products of hydrogenic orbitals with either two-electron, one-center or one-electron, two-center first-order wave functions. These first-order wave functions are obtained from calculations on He-like and H1 2 -like systems. Similarly, the nth-order wave function decouples so that the most complex terms are just the nth-order wave functions of all the p-electron, q-center subsystems ( p1q5n12) contained in the molecule. We illustrate applications of this method with some results, complete through third order in the energy, for H1 3 -like molecules. These are compared with accurate variational results available in the literature. We conclude that, through this order, this perturbation approach is capable of yielding results comparable in accuracy to variational calculations of moderate complexity. The ease and efficiency with which such results can be obtained suggests that this method would be useful for generating detailed potential energy surfaces for polyatomic molecules. © 1995 American Institute of Physics. I. INTRODUCTION Z-dependent perturbation theory ~ZDPT! has long been a powerful computational tool for atomic systems. Little comparable work has been done, however, in molecular systems. Examples include the work of Goodisman1 and of Matcha and Byers Brown2 ~on diatomic systems! based on an unperturbed Hamiltonian taken as the sum of one-electron, diatomic ~H1 2 -like! Hamiltonians, and the work of Dvorácek and Horák3 ~on the hydrogen molecule!, of Chisholm and Lodge,4 ~on two-electron diatomic systems!, and of Montgomery, Bruner, and Knight5 ~on ten-electron hydrides!, all of which utilize single-center, hydrogenic Hamiltonians to describe the unperturbed system. These approaches illustrate one characteristic of the application of ZDPT to molecules; it is possible to construct a variety of unperturbed Hamiltonians and still retain the essential features of the theory. The advantages of ZDPT and a multiperturbation approach for atomic systems are discussed in Sanders.6 The present paper extends this earlier work to polyatomic systems. One of the advantages of ZDPT is its ability to provide results for an entire isoelectronic sequence from a single calculation. While this is less significant for polyatomic systems, where few molecular isoelectronic sequences have more than one or two physically interesting examples, the a! Present address: Department of Physics, An Najah University, P.O. Box 7, Nablus, West Bank via Israel. J. Chem. Phys. 102 (12), 22 March 1995 method retains some important advantages for molecular systems. As usual, the inverse nuclear charge appears as a natural perturbation parameter of the method. In addition, the ratio of the nuclear charges also appears as a natural choice of expansion parameter in a multiple perturbation theory. These ensure a rapid convergence of the multiperturbation series for molecules possessing at least one heavy atom. Of greater importance, with these perturbation parameters the individual multiperturbation wave functions and energy coefficients are independent of the nuclear charges as well as of the total number of electrons and the overall electronic configuration of the system. Consequently, results obtained for small systems can be transferred without modification to larger systems which contain the electron configuration of the smaller system as a subconfiguration. It is these characteristics of the method that suggest that ZDPT can be an accurate and efficient method for studying potential energy surfaces of polyatomic systems. Chisholm and Lodge4 studied the ground state of twoelectron diatomic systems ~specifically H2 and HeH1! through second-order in the energy. The present paper extends these calculations to third order in the energy. These 12 results are then incorporated into a study of the H1 3 , HeH2 sequence as the simplest prototype of a polyatomic molecule. As in Refs. 3–5, we place all the unperturbed ~hydrogenic! orbitals on the same nuclear charge. Of the various possible choices of a zero-order Hamiltonian, this produces the simplest form for the higher-order perturbation coefficients. It 0021-9606/95/102(12)/4919/12/$6.00 © 1995 American Institute of Physics 4919 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 4920 Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules also means that contributions of the interelectron interaction can be obtained directly from ZDPT calculations for atoms. The complete, first-order wave function of any molecule can then be constructed from the first-order, atomic ~He-like! pair functions of all two-electron configurations present in the zero-order wave function, together with the first-order, oneelectron diatomic ~H1 2 -like! wave functions of all the orbitals in the zero-order wave function. Similarly, the second-order correction to the energy consists entirely of one-center threeelectron, two-center two-electron, and three-center oneelectron contributions, no matter how complex the molecule. Continuing to higher-order, the maximum degree of complexity of the calculation increases in a predictable manner, with each additional order of the calculation introducing either an additional electron or an additional center to the expansion coefficients. Despite its simplicity, this zero-order Hamiltonian is clearly not the best choice for many polyatomic molecules, particularly homonuclear molecules. Nevertheless, it will serve here to illustrate the general structure of the multiperturbation expansion and the general characteristics of the method. A more natural choice of zero-order Hamiltonian would distribute the electrons for the system among the nuclear centers. This would significantly improve the initial, unperturbed electron density and hence also improve the convergence of the perturbation series, particularly as one approaches the separated-atom limit. Of course, this would also increase the complexity of the calculation at each order of the perturbation by precisely the number of additional centers that have been introduced into the zero-order Hamiltonian. Such generalizations of the method are straightforward and will be examined in more detail in a later paper. We describe an unperturbed N-electron, M -center polyatomic molecule as N ‘‘hydrogenic’’ electrons on a single center of charge Z A . In charge-scaled atomic units,7 this Hamiltonian is written as H5H 0 1H 1 , where H 05 ( i51 N H 15 S S 2 H i1j 5 1 , rij D 1 1 D2 , 2 i r iA N ( ( i51 G 0 c 0 50, ~3! G 0 c i1j 1G i1j c 0 50, ~4! G 0 c i1a 1G i1a c 0 50, ~5! G 0 c i2j 1G i1j c i1j 2 e i2j c 0 50, ~6! G 0 c i2a 1G i1a c i1a 2 e i2a c 0 50, ~7! j, jk j, jk G 0 c i1,1 1G i1j c 1jk 1G 1jk c i1j 2 e i1,1 c 0 50, ~8! j,i a G 0 c i1,1 1G i1j c i1a 1G i1a c i1j 2 e i11j,i a c 0 50, ~9! a ,i b a ,i b G 0 c i1,1 1G i1a c i1b 1G i1b c i1a 2 e i1,1 c 0 50, ~10! G a1 5H a1 2 e a1 . where G 0 5H 0 2 e 0 , and Equations ~4! and ~6! reduce to two-electron, one-center equations, while Eqs. ~5! and ~7! simplify to one-electron, two-center equations. Equations ~8!, ~9!, and ~10! are three-electron one-center, twoelectron two-center, and one-electron three-center equations, respectively. From these perturbation differential equations one can obtain all corrections to the energy through fifth order.8 Expressions for the third-order energy coefficients are presented below. In these expressions, a, b, and c represent any one of the perturbations i j or i a , with the restriction that all perturbations appearing in a coefficient must be different: e a1 5 ^ c 0 u H a1 u c 0 & , e a2 5 ^ c a1 u G a1 u c 0 & , II. THEORY N any case, it is clear that the convergence of portions of the multiperturbation series will be slower than might be anticipated based on the size of Z A alone. Treating each term in Eq. ~2! as a separate perturbation, we obtain the multiperturbation differential equations.6 Through second order, these are ~1! M l i j H i1j 1 j.i H i1a 52 ( a 5B 1 , r ia l i a H i1a D a b e a,b 1,1 52 ^ c 1 u G 1 u c 0 & , e a3 5 ^ c a1 u G a1 u c a1 & 22 e a2 ^ c a1 u c 0 & , ~11! a a b a b a a b e a,b 2,1 52 ^ c 1 u G 1 u c 1 & 1 ^ c 1 u G 1 u c 1 & 22 e 2 ^ c 1 u c 0 & a 22 e a,b 1,1 ^ c 1 u c 0 & , a b c b a c a c b e a,b,c 1,1,1 52 ~ ^ c 1 u G 1 u c 1 & 1 ^ c 1 u G 1 u c 1 & 1 ^ c 1 u G 1 u c 1 & c a,c b b,c a 2 e a,b 1,1 ^ c 1 u c 0 & 2 e 1,1 ^ c 1 u c 0 & 2 e 1,1 ^ c 1 u c 0 & ! . , ~2! 21 where l i j 5Z 21 A and l i a 5Z a Z A ; the Z a being the charges of the other atoms in the molecule. The perturbation expansion coefficients are then independent of the nuclear charges and completely transferable from one system to another. This choice of zero-order Hamiltonian also reduces to a minimum the number of nuclear centers that can appear at any particular order. Note that the ratio of the nuclear charges appears in these perturbation expansions in addition to the usual inverse of the nuclear charge. Obviously, Z A should be chosen as the largest of the nuclear charges in the molecule, if possible. In Extensions to higher order are straightforward, and examples can be found in Ref. 6. It can be shown that all ‘‘unlinked’’ wave functions decouple into simple products of their ‘‘linked’’ components, e.g., 12 34 c 12,34 1,1 5 c 1 c 1 , a ,2 a c 11,1 5 c 11 a c 21 a , ~12! 4a a c 12,23,4 5 c 12,23 1,1,1 1,1 c 1 . From this it follows that all such ‘‘unlinked’’ energy coeffia cients disappear, e.g., e12,3 1,1 50. These results ensure that at nth order the wave functions and energy coefficients cannot involve more than p electrons and q centers, where p1q5n J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules 12. This in turn means that an nth order perturbation correction can involve no more than n11 coupled electrons while the greatest number of nuclear centers involved in nth order is also n11. A. Charge-scaling in molecules In contrast to the corresponding expressions for atoms, the multiperturbation expansions for a molecule have a much more complex behavior with respect to the nuclear charges. The multiperturbation expansion for the electronic energy of a heteronuclear diatomic molecule serves to illustrate this behavior. In charge-scaled atomic units, this is E5 e 0 1 ( S l nB e nM 1Z 21 e A1 1 A n51 S 1Z 22 e A2 1 A 1••• , l B 5Z B Z 21 A . ( n51 ( l nB e A,M 1,n n51 D S l nB e A,M 1Z 23 e A3 1 2,n A D ( n51 l nB e A,M 3,n D ~13! where A comparable expression for the energy of a polyatomic molecule requires a simple and obvious extension of this form and introduces no new behavior with respect to charge-scaling. The notation in Eq. ~13! has been simplified so that each ‘‘molecular’’ coefficient, enM , each ‘‘atomic’’ coefficient, eAn , and all ‘‘mixed’’ coefficients have implicit in them sums over all N electrons of the molecule. The first two terms in this expression represent the sum of the energies of all states of those one-electron diatomic molecules which are subsystems of the molecule, while the sum over all the first ~‘‘atomic’’! coefficients in each of the bracketed terms represents the energy of the atom which is a subsystem of the diatomic molecule. Since l B >Z 21 A , the convergence of those parts of the series involving lB can be slower than that of the purely ‘‘atomic’’ contributions. An obvious exception to this occurs for the hydrides, where all the multiperturbation expansion parameters are identically Z 21 A , and the multiperturbation expansion simply becomes a formal device for identifying the contributions from subsystems of the molecule. Perturbation energies and other properties of hydrides will thus have simple charge-scaling behavior. At the other extreme are the homonuclear molecules, for which all lB 51. Here, the results obtained via perturbation theory can be seriously affected by a premature truncation of the perturbation series. Note in particular that the entire portion of the energy arising from purely one-electron diatomic energy coefficients is of the same order ~in terms of the nuclear charges! as the zeroorder term. Similarly, terms of a particular order in Z 21 A will contain contributions from multiperturbation coefficients of much higher nominal order. Hence, for homonuclear molecules, truncating the perturbation sum at some order can yield a relative error which actually increases as Z A increases. The convergence of the energy can be improved considerably, however, by including the higher-order coefficients of one-electron diatomics. This is effectively what was done in the work of Joulakian9 on isoelectronic homonuclear diatomics. 4921 Finally, it must be kept in mind that all lengths, including internuclear separations, have been scaled by Z A . Hence, energies calculated at a particular scaled internuclear distance will actually correspond to different internuclear separations for different members of an isoelectronic molecular sequence according to the relation R~bohr!5Z 21 A 3R~scaled a.u.!. B. Degeneracy In degenerate multiple perturbation theory, the multiperturbation expansion becomes algebraically more complex.8 Nevertheless, the multiperturbation expansion coefficients can still be constructed from the corresponding coefficients of smaller subsystems of the molecule. Chisholm and Lodge10 have described one approach to this problem, demonstrating how these perturbation coefficients can be constructed. For the ground state of two-electron molecules, the zeroorder wave function is a product of hydrogenic 1s orbitals and is not degenerate. Thus, in what follows, this system will serve to illustrate the structure of the method most clearly and simply. The single-center, zero-order wave functions for excited states of these systems and for the ground state of three-electron molecules will be degenerate, however. To illustrate the application of the method to these more complex, often degenerate systems, a brief discussion of the multiperturbation expressions for H3-like molecules is presented in the Appendix. III. APPLICATION TO H1 3 -LIKE MOLECULES For two-electron, three-center molecules, the complete perturbation is H 15 S F D S 1 1 1 1 1 1 2Z B 1 2Z c 1 Z A r 12 r 1B r 2B r 1C r 2C DG . ~14! For Z C 50, this becomes the perturbation for H2-like molecules. Hence, in what follows, all expressions are written for H1 3 -like systems, expressions for two-electron diatomics being obtained by simply setting Z C equal to zero. The complete, first-order energy coefficient for the ground state of the molecule is 1 1s s s e 1 5 e 11 S 12Z B e 1s 1 ~ R B ! 12Z C e 1 ~ R C ! . ~15! 11S Here e 1 5 85 is the first-order correction to the ground state energy of a two-electron atom ~and corresponds to an ei1j ! while s e 1s 1 ~ R ! 52 S D 1 1 1 11 e 22R R R ~16! is the first-order correction to the ground state energy of a one-electron diatomic molecule with internuclear distance R ~and corresponds to an ei1a !. R B and R C are the internuclear distances between the charge Z A and the two perturbing charges, Z B and Z C . The angle subtended by these two charges is denoted by Q. J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 4922 Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules Each term in the perturbation, Eq. ~3!, yields a corresponding term in the complete first-order wave function 1 2 1s s s e 2 5 e 12 S 12Z 2B e 1s 2 ~ R B ! 12Z C e 2 ~ R C ! 1 s ,1s s 12Z B Z C e 1s ~ R B ,R C ,Q ! , 1,1 1s s s 11s ~ r1 ! c 1s 1 ~ R B ,r2 !# 1Z C @ c 1 ~ R C ,r1 ! 1s ~ r2 ! s 11s ~ r1 ! c 1s 1 ~ R C ,r2 !# . ~17! 1 c 11 S can be obtained accurately from variational perturbation s can be expressed in closed form in calculations11 while c1s 1 confocal elliptic coordinates,12 but is here also obtained from a variational perturbation calculation. With this wave function, the second- and third-order energy coefficients can be calculated. The second-order energy is given by 1 1 1Z B e 11,1S,1s s ~ R B ! 1Z C e 11,1S,1s s ~ R C ! 1 s c 1 5 c 11 S ~ r1 ,r2 ! 1Z B @ c 1s 1 ~ R B ,r1 ! 1s ~ r2 ! 1 ~18! 1 s where e 12 S 5 20.157 666 4,11 e1s is known exactly,12 2 1 1 S,1s s and e 1,1 is given in Ref. 4. Note that the dependence of the electronic energy on the angle Q makes its first appearance at this order through the three-center term, K U s ,1s s s e 1s 52 c 1s 1,1 1 ~ R B ,r1 ! 2 L U 1 1s ~ r1 ! . r 1C ~19! The third-order energy is given by 1 1 1 3 1s s 1 S,1s s s e 3 5 e 13 S 12Z 3B e 1s ~ R B ! 1Z C e 12,1S,1s s ~ R C ! 1Z 2B e 11,2S,1s s ~ R B ! 1Z 2C e 11,2S,1s s ~ R C ! 3 ~ R B ! 12Z C e 3 ~ R C ! 1Z B e 2,1 1 s ,1s s s ,1s s S,1s s ,1s s 12Z 2B Z C e 1s ~ R B ,R C ,Q ! 12Z B Z 2C e 1s ~ R B ,R C ,Q ! 1Z B Z C e 11,1,1 ~ R B ,R C ,Q ! . 2,1 1,2 All of the singly-subscripted coefficients above are either known exactly13 or are known to high precision from variational perturbation calculations.11 The remaining multiperturbation coefficients can be computed, via Eqs. ~11!, from the appropriate components of the first-order wave function. The total energy through nth order, in atomic units, is given by ~20! c 11 S ~ r1 ,r2 ! 5 ~ 11 P 12 ! ( c nml r n1 r m2 P l ~ cos u 12 ! 1 nml [email protected] 2 b l ~ r 1 1r 2 !# , n s c 1s 1 ~ R,ri ! 5 ( c nl r i P l ~ cos u i ! exp~ 2 b 8 l ri!. ~23! ~24! nl n E n ~ R B ,R C ,Q ! 5Z 2A ( p Z2 A e p1 p50 1 Z AZ B Z AZ C 1 RB RC Z BZ C . R BC ~21! The total wave function, truncated through first order, can also be used to obtain a variational bound on the energy E v 5Z 2A e 0 1Z A e 1 1 1 Z BZ C . R BC e 2 1 e 3 /Z A 11Z 22 A ^ c 1u c 1& 1 Z AZ B Z AZ C 1 RB RC ~22! Results for both the third-order perturbation sums and this variational energy for H1 3 -like systems are presented in the tables, where they are compared with accurate, variationally obtained energies. IV. METHOD The first-order wave functions required by the method have been obtained variationally. To simplify the calculation of the multicenter integrals that appear in the energy expansion coefficients, we have used single-center basis sets for all wave functions. Hence, For the ‘‘atomic’’ wave function, Eq. ~23!, all terms with l<16 and n1m12l<20 were included for a total of 501 terms. For the ‘‘molecular’’ wave function, Eq. ~24!, all 221 terms with l<16 and l1n<20 were utilized. For both wave functions, the nonlinear parameters, bl , of each partial wave were separately optimized. Once obtained, the optimized ‘‘atomic’’ wave function is stored. The nonlinear parameters for the ‘‘molecular’’ wave function, however, must be optimized for each value of the internuclear distance, R. For the sake of efficiency, these parameters were obtained at intervals of 0.10 bohr ~and at smaller intervals near the equilibrium distances!. These were then used to interpolate for the parameters at other values of R. Since the calculation of these one-electron, ‘‘molecular’’ wave functions is extremely rapid, it costs little in efficiency to simply recalculate them as needed. Hence, none of these ‘‘molecular’’ wave functions were stored and only the interpolation table for the nonlinear parameters was saved. With these wave functions in hand, it only remains to calculate the energy expansion coefficients for each particular choice of R B , R C , and Q. These can be calculated very efficiently; for each such point, the entire calculation of the energy through third order for all values of Z of interest consumes about 20 s of cpu time on an IBM 9021 vector processor with 2 cpu’s. Hence, to calculate all the data actually presented in the tables to follow requires less than 20 min of cpu time. Even greater efficiency is possible in cal- J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules 4923 TABLE I. Some perturbation energy coefficients ~in a.u.!; a comparison of the present results with exact values.a R ~bohr! s b 2 e 1s 2 s c 2 e 1s 3 e 11,1S,1s s d s ,1s s e 1s 1,1 0.40 0.386 56 0.386 57 0.312 05 0.312 08 0.249 18 0.249 21 0.199 27 0.199 31 0.160 29 0.160 33 0.070 19 0.070 25 0.025 67 0.025 69 0.009 769 0.009 774 20.022 30 20.022 29 20.017 50 20.017 44 20.005 93 20.005 86 0.005 97 0.005 99 0.015 08 0.015 17 0.023 14 0.023 25 0.011 02 0.011 07 0.003 59 0.003 61 0.529 45 0.529 60 0.448 98 0.449 10 0.367 98 0.368 11 0.293 88 0.294 00 0.229 98 0.230 08 0.075 03 0.075 09 0.015 36 0.015 39 0.002 81 0.002 82 20.583 15 20.583 17 20.339 32 20.339 37 20.160 36 20.160 37 20.044 58 20.044 60 0.022 67 0.022 68 0.068 73 0.068 74 0.032 972 0.032 974 0.013 910 0.013 911 0.60 0.80 1.00 1.20 2.00 3.00 4.00 1 e a Each first entry below corresponds to the present results. The second entry in this column corresponds to the results of Ref. 12. c The second entry in this column corresponds to the results of Ref. 13. d The second entry in this column corresponds to the results of Ref. 4~b!. e These results are for Q5180°, the second entry in this column corresponding to the results of Ref. 14. b culating the potential energy surfaces presented in the figures. For each value of R B , R C , the energies for all Q of interest are calculated together. Since all coefficients that do not depend on Q are not recalculated, less than 10 s of cpu time is required for each point. Additional savings in computational time are possible whenever R B 5R C so that many of the multiperturbation coefficients are identical. s s and e1s In Table I we compare exact values12,13 of e1s 2 3 1s s with those obtained with our approximate c1 and find that the approximate coefficients are accurate to at least four decimal places over the entire range of internuclear distances considered here. The simple wave function utilized here for 1 1 1 c 11 S yields values for e 12 S and e 13 S which agree with accurate values of these coefficients to three decimal places. In Table I we also compare calculated and exact values4 of 1 e 11,1S,1s s . Here the results obtained with the ‘‘atomic’’ firstorder wave function are accurate to four decimal places, except for R<1, where they are in error by roughly one unit in the fourth decimal place. ~These are obtained with greater s accuracy if calculated with c1s 1 .! Comparing exact values of 1s s ,1s s e1,1 ~Ref. 14! with those calculated with the approximate first-order ‘‘molecular’’ wave function shows that the latter agree with the exact values to four or five decimal places. Not surprisingly, the approximate one-electron ‘‘molecular’’ first-order wave functions are more accurate than the twoelectron ‘‘atomic’’ wave functions, but between them, they appear capable of calculating the ‘‘mixed’’ perturbation energy coefficients to roughly four decimal places for the range of R considered here. Note that, in calculating the total energies, only exact values of these coefficients have been used where they are available. Hence, the total energies presented in the tables cannot be expected to agree with accurate variational energies to more than four decimal places. Conversely, if the present calculations agree with accurate variational en- FIG. 1. Energy for H2 ; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ; ~!! E v ; ~s! variational results of Ref. 15. ergies by less than about four decimal places, the discrepancy can be attributed to the truncation of the perturbation series. V. RESULTS AND DISCUSSION Although the main focus of this work is the calculation of polyatomic molecular energies, a brief examination of results for diatomic molecules will illustrate the behavior of the multiperturbation series in the present method. A. Diatomic molecules: H2 A comparison of the present perturbation results with accurate variational calculations15 for the ground state of the H2 molecule is presented in Fig. 1. In this figure, the convergence of the perturbation results can be gauged by examining the behavior of successive truncated perturbation sums, the E n of Eq. ~21!, for the total energy of the molecule. These results indicate that the variational energy, E v , obtained with the first-order wave function is consistently less accurate than the corresponding truncated third-order energy sum, E 3 , obtained with the same wave function. This is not unexpected, as variational expressions based on perturbation wave functions such as that of Eq. ~22! seem to consistently yield poorer results than the corresponding truncated perturbation series. In fact, in this instance at least, E v is less accurate than E 2 . It is also interesting to note that E 2 is quite similar to the Hartree–Fock result16 for this molecule. The present calculations have been carried out in sufficient detail to verify that all the E n and E v have a minimum in the potential energy close to the equilibrium internuclear distance for this molecule. E 3 yields the best result at R51.36 bohr, within 0.04 bohr of the correct value of 1.40 bohr. Figure 2 displays differences between the ‘‘exact’’ variational results and the perturbation results, J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 4924 Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules FIG. 2. Truncation error, Dn , for H2 ; ~n! D1 ; ~L! D2 ; ~h! D3 ; ~!! E v . D n 5E n 2E exact , showing that the perturbation results are most accurate in the vicinity of the energy minimum. For example, the smallest error in E 3 occurs at R51.2 bohr where it is 0.006 a.u., or 0.5%. For larger R, all the perturbation results become increasingly less accurate, so that by R53 the error is approximately 0.1 a.u. By R55 the error for E 2 , E 3 , and E v has grown to '0.25 a.u., a relative error of just over 25%. The present method fails to give accurate results for large R for this molecule since the zero-order wave function tends to an incorrect separated-atom limit ~H21H1!. H2 is, of course, a particularly difficult case for the Z 21 expansion. Nevertheless, the convergence of the perturbation expansion for this particular molecule is quite satisfactory. For this system, e3 is negative for the entire range of R presented in the table except for the smallest R and it provides a considerable improvement over the second-order results of Chisholm and Lodge,4 reducing the error in the energy by almost an order of magnitude. In fact, the present results show that the relative error for E 3 remains under 1% for 1&R&2 bohr. In general, use of similar zero-order wave functions in homonuclear molecules should produce perturbation series which converge most rapidly for intermediate values of the internuclear distance, but become suspect for larger values of R. FIG. 3. Energy for HeH1; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ; ~!! E v ; ~s! variational results of Ref. 17. at the equilibrium distance. Indeed, E 3 does not yield quite as good an equilibrium distance as either E 2 or E v ; R51.41 as compared to R51.50 and 1.43, respectively. All however lie within 0.05 bohr of the correct value of R51.463 ~Ref. 17!. In contrast to the other systems studied here, E v is consistently better than both E 2 and the Hartree–Fock energy18 for R&2. Again, as in the case of H2 , E 2 and the Hartree–Fock energy are quite similar for R&2, beyond which point E 2 is a significant improvement over E v . Since the perturbation series for this molecule converges more rapidly, the improvement that E 3 provides over E 2 is not as large here as in the case of H2 . Nevertheless, E 3 generally reduces the error relative to accurate variational results17 to less than half what it was for E 2 . Over the entire range of R, the relative error in E 3 is less than 1% for this molecule. Indeed, it is less than 0.5% for 1.0*R*2.5. B. Diatomic molecules: HeH1 Figures 3 and 4 present results for HeH1 in a form similar to those for H2 . For this molecular ion, all perturbation parameters are equal to 21 and a more rapid convergence of the perturbation series can be expected. Moreover, unlike the case of H2 , the zero-order wave function does represent the separated-atom limit correctly. Thus, for large R, the perturbation results are very well converged except for E 1 , where 1 the discrepancy is primarily due to the absence of e 12 S . For this system, we find that E 2 and E 3 give slightly better results at both larger and smaller internuclear distances rather than FIG. 4. Truncation error, Dn , for HeH1; ~n! D1 ; ~L! D2 ; ~h! D3 ; ~!! E v . J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules 4925 TABLE II. Total energies ~in a.u.! for the H1 3 molecule; the first-order variational energy, E v , and truncated energy sums, E n , compared with variational energies. D 3h Symmetry R ~bohr! 1.137 1.250 1.277 1.414 1.549 1.644 1.650 1.656 1.819 1.900 2.050 2.094 2.200 2.235 2.750 3.052 3.350 4.500 5.500 7.000 R ~bohr! 1.130 1.257 1.431 1.435 1.438 1.452 1.462 1.507 1.509 1.537 1.571 1.609 1.619 1.695 1.743 1.756 1.756 1.794 1.848 1.894 1.962 1.995 2.085 2.154 2.277 2.419 3.092 4.067 5.054 6.044 8.031 10.025 a Ev E1 E2 E3 E~variational!a 20.913 21.015 21.033 21.105 21.145 21.159 21.160 21.161 21.166 21.163 21.150 21.145 21.130 21.125 21.032 20.975 20.923 20.776 20.706 20.649 20.481 20.584 20.603 20.678 20.724 20.743 20.744 20.745 20.762 20.765 20.764 20.763 20.758 20.756 20.716 20.691 20.667 20.597 20.557 20.518 21.265 21.282 21.283 21.280 21.265 21.249 21.248 21.247 21.215 21.197 21.163 21.153 21.128 21.120 21.009 20.954 20.907 20.787 20.730 20.681 21.230 21.304 21.317 21.365 21.387 21.391 21.391 21.391 21.381 21.370 21.343 21.334 21.310 21.302 21.166 21.088 21.018 20.831 20.746 20.682 21.233 78 21.286 40 21.295 48 21.327 44 21.341 24 21.343 79 21.343 83 21.343 78 21.338 33 21.331 39b 21.319 13 21.314 56 21.301 51b 21.298 63 21.234 12b 21.198 43b 21.166 32b 21.076 51b 21.036 26b 21.011 21b Q 124.141 117.047 114.447 109.417 132.515 99.513 95.863 82.960 157.418 76.835 70.958 65.344 103.498 54.929 50.129 20.043 99.957 45.594 41.316 23.069 33.488 31.510 26.551 68.199 29.448 31.275 75.964 79.588 81.637 83.063 84.928 85.939 Ev 21.041 21.117 21.174 21.174 21.180 21.175 21.176 21.179 21.195 21.179 21.176 21.170 21.194 21.165 21.166 20.353 21.193 21.164 21.156 20.872 21.122 21.107 21.050 21.175 21.130 21.153 21.138 21.114 21.104 21.102 21.102 21.101 C 2 v Symmetry E1 20.857 20.918 20.969 20.957 21.004 20.931 20.920 20.875 21.037 20.847 20.817 20.783 20.963 20.764 20.781 0.310 20.955 20.795 20.805 20.606 20.807 20.802 20.771 20.828 20.840 20.859 20.870 20.877 20.878 20.878 20.880 20.880 E2 21.307 21.313 21.298 21.305 21.270 21.315 21.317 21.316 21.235 21.309 21.295 21.275 21.283 21.262 21.273 20.320 21.261 21.281 21.284 21.150 21.274 21.266 21.230 21.229 21.244 21.234 21.175 21.143 21.129 21.123 21.122 21.121 E3 21.131 21.205 21.254 21.262 21.239 21.282 21.291 21.328 21.237 21.348 21.365 21.380 21.285 21.385 21.375 20.784 21.287 21.361 21.341 21.029 21.283 21.263 21.193 21.343 21.265 21.285 21.243 21.195 21.177 21.170 21.167 21.166 E~variational!a 21.238 38 21.278 81 21.301 07 21.305 22 21.289 43 21.314 26 21.317 79 21.330 28 21.281 33 21.335 63 21.339 92 21.342 76 21.309 96 21.342 62 21.338 81 20.889 35 21.304 81 21.331 83 21.321 02 21.086 63 21.284 03 21.269 23 21.215 80 21.316 90b 21.267 52 21.280 59 21.245 92b 21.201 25b 21.182 25b 21.176 28b 21.174 04b 21.173 45b Unless otherwise noted, these are all from Frye et al., Ref. 19. Talbi and Saxon, Ref. 20. b C. Triatomic molecules: H1 3 As in the case of H2 , this molecule presents a severe test for the method. In addition to slow convergence, the calculated ground state energies also have the same difficulties at the separated-atom limit noted for H2 . Table II presents results for H1 3 in D 3h ~equilateral triangle! and C 2 v ~isosceles triangle! symmetry. In addition, Figs. 5, 6, and 7 present potential energy surfaces based on E 1 , E 2 , and E 3 for C 2 v J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 4926 Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules FIG. 5. E 1 potential energy surface for H1 3 ; contours are 0.05 a.u. apart starting at 21.19 a.u. FIG. 7. E 3 potential energy surface for H1 3 ; contours are 0.05 a.u. apart starting at 21.39 a.u. symmetry. In Fig. 5, we see that E 1 does not reproduce the correct equilibrium separation for D 3h symmetry. Surprisingly, however, it does yield an equilibrium distance of R51.62 at the minimum of its potential energy surface, close to the correct value of R51.65, but this occurs for a linear geometry. Note that the only dependence on Q at this order is through the nuclear potential, Z B Z C /R BC . Adding the second-order energy introduces the first Q-dependent contribution to the electronic energy. The overall second-order coefficient is everywhere negative and becomes more so for small R and Q. This places the minimum of the potential in Fig. 6 at R51.20, Q590°, shifting the minimum in the right direction but by too much in R and not enough in Q. The third-order energy coefficient itself has a minimum very close to the correct equilibrium distance for the molecule, but at Q50°. As a consequence, this coefficient produces a minimum in the potential energy very close to the correct equilibrium separation but slightly too low in Q. The potential energy surface of Fig. 7 shows a minimum energy at R51.66 and Q554°. In contrast, the potential produced by E v and displayed in Fig. 8 shows a long trough with a shallow secondary minimum at its head near Q590° and the minimum at its base at Q5180°. Of course the energy for this minimum lies well above those of E 2 and E 3 . A number of variational calculations have been carried out to an accuracy of a few thousandths a.u. or better for this molecule. Notable among these are the calculations of Frye et al.,19 who obtain very accurate results over a range of internuclear separations and geometries. Total energies for larger internuclear distances can be found in Talbi and Saxon.20 The accuracy of these calculations permit an unambiguous assessment of the convergence of the perturbation results. A comparison of these results for D 3h and C 2 v sym- FIG. 6. E 2 potential energy surface for H1 3 ; contours are 0.05 a.u. apart starting at 21.21 a.u. FIG. 8. E v potential energy surface for H1 3 ; contours are 0.05 a.u. apart starting at 21.19 a.u. J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules FIG. 9. Truncation error, D3, for H1 3 ; ~n! Q,50°; ~L! 50°,Q,70°; ~s! 70°,Q,90°; ~h! 90°,Q,120°; ~!!Q.120°. metries is presented in Table II. Figure 9 displays the differences between the variational values and E 3 , grouped according to the value of Q. In the vicinity of the equilibrium distance, most points fall in the range of 60.05 a.u., a relative error of about 3.5%. Exceptions to this occur for those points with both the largest and the smallest values of Q; i.e., all those with Q.120° and two points belonging to the group with Q,50° that actually have the smallest values of Q~'20°!. These all have relative errors ranging from 5% to 10%. At larger internuclear distances, points with Q560° show an error which increases with R, while points which correspond to Q'90° have an error which remains fairly constant at about 0.006 a.u. It is clear that much of this behavior is determined by the initial charge distribution imposed by the zero-order function. Thus, those points with R.3 and Q'90°, all of which have geometries with one internuclear distance held close to the equilibrium separation for H2 ,20 have errors which are quite close to that of the H2 molecule at equilibrium, the perturbation coefficients involving large internuclear separations contributing little to the total energy. Conversely, those points at large R corresponding to an equilateral geometry all have errors which increase with R in a manner similar to the results for H2 at large R. 4927 FIG. 10. Energy for equilateral H1 3 ; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ; ~!! E v ; ~s! variational results, Ref. 17. steadily for increasing R, reaching 5% in the vicinity of R'2.7 and 10% by R'3. However, for 1,R,2.5, the relative error is always less than 3.5%. Both E 3 and E v ~but not E 2! predict a minimum in the potential energy for this symmetry at the correct internuclear distance to within 60.01 bohr. Surprisingly, E 3 is more negative than the ‘‘exact’’ variational values for 1.25,R,2.5, reaching its greatest discrepancy in the vicinity of the equilibrium position. However, as in the case of H2 , these results show a satisfactory convergence and a significant improvement in the total energy over the second-order results. For nonsymmetric geometries of this molecule, comparison to the results of Ref. 19 shows that the truncation error is generally about 0.02–0.05 a.u. for the internuclear distances considered there, with the range from largest to smallest being from 0.006 to 0.09 a.u. D. Equilateral H1 3 For the equilateral geometry, a comparison to H2 is useful throughout the entire range of R. Figures 10 and 11 present the results for equilateral H1 3 in a manner similar to Figs. 1 and 2 for H2 . Comparison of Figs. 2 and 11 indicates that, in this case, the errors for E n in H1 3 behave very much like the corresponding errors in H2 , although these are, in general, considerably smaller for H2 . For example, for R in the range 1.2,R,1.8, errors in H1 3 are roughly 2 to 5 times larger than those found for H2 at similar internuclear distances. Ultimately, and again as in the case of H2 , as R increases these results converge more slowly and to the wrong limiting value. Thus, the relative error in E 3 increases FIG. 11. Truncation error, Dn , for equilateral H1 3 ; ~n! D1 ; ~L! D2 ; ~h! D3 ; ~!! E v . J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 4928 Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules TABLE III. Total energies ~in a.u.! for the linear, symmetric HeH11 molecule; the first-order variational energy, 2 E v , and truncated energy sums, E n , compared with the variational results. R ~bohr! 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Ev E1 E2 E3 E~variational!a 22.188 22.552 22.653 22.700 22.730 22.753 22.770 22.784 22.795 22.030 22.390 22.497 22.550 22.583 22.607 22.625 22.639 22.650 22.181 22.554 22.660 22.710 22.742 22.765 22.783 22.797 22.808 22.197 22.564 22.664 22.710 22.739 22.762 22.779 22.792 22.803 22.155 91 22.527 73 22.625 87 22.663 36 22.683 70 22.698 66 22.711 56 22.722 94 22.732 74 a Zetik and Poshusta, Ref. 21. E. Triatomic molecules: HeH12 2 Comprehensive variational calculations for this unstable system over a range of internuclear distances can be found in Zetik and Poshusta21 who examined the potential energy surface for a linear geometry. These results are compared with the present calculations in Table III and Fig. 12 for the linear, symmetric molecule. This comparison shows that E 2 , E 3 , and E v all lie below this variational calculation by roughly 0.05 a.u. Since E v is itself an upper bound to the exact energy, it is clear that all of these results represent an improvement over the variational result. Note that both E 2 and E 3 lie lower in energy than E v for R>1, with E 2 being the more negative for R.2.5. The absolute accuracy of these results is difficult to judge in the absence of more accurate variational calculations. Recalling the comparable behavior of the perturbation series for H1 3 and H2 noted earlier, a corresponding and HeH1 series might be similarity between the HeH11 2 expected. Indeed, the E n of Table III do suggest a rapid convergence of the perturbation series for HeH11 2 . In particular, e3 for this system is virtually identical to e3 for HeH1 once R exceeds about 2 bohr, indicating that contributions from multiperturbation coefficients with two ‘‘molecular’’ FIG. 12. Energy for linear, symmetric HeH11 2 ; ~n! E 1 ; ~L! E 2 ; ~h! E 3 ; ~!! E v ; ~s! variational results of Ref. 21. perturbations, as appear in polyatomic systems, are quite small for large R. In fact, the only significant contribution to 1 e3 in this range of R is from e 13 S . In this context, it should be remembered that internuclear distances are scaled by the nuclear charge, Z A , so that these energy coefficients are actually calculated with values of R that are twice their nominal value in bohr. Similar improvements in convergence can be expected for molecules containing heavier atoms than those appearing in these two-electron systems. This should be particularly true for heteronuclear molecules, especially those with a single heavy ion. For homonuclear molecules, where the multiperturbation parameter lM 51, the relative error ~in charge-scaled atomic units! can actually increase with Z along a particular isoelectronic sequence, while the absolute error remains roughly constant. In nonscaled atomic units, M . This error is easily the absolute error increases as Z 2A e n11 corrected, however, by simply adding the ‘‘missing’’ portion of the electronic energy of the appropriate states of the oneelectron diatomic molecule. VI. SUMMARY In Z-dependent perturbation theory, the lowest-order wave functions for a polyatomic molecule are not only independent of the nuclear charges, but are also independent of the total number of nuclear centers and electrons in the molecule. The complexity of the problem is then determined by the highest order of the calculation. With the present choice of H 0 , the first-order wave function for any polyatomic molecule is described completely in terms of two-electron, onecenter ~atomic! and one-electron, two-center ~molecular! first-order wave functions. These are separately obtained from calculations on He-like and H1 2 -like systems. At nthorder the wave function for a polyatomic molecule decouples into a sum over the nth-order wave functions of all p-electron, q-center subsystems (p1q5n12) that are contained within the molecule of interest. We have illustrated the application of this method with some results, complete through third order in the energy, for H1 3 -like molecules. In applying this method, we have chosen to describe the system by a zero-order wave function that minimizes the complexity of the calculations. The perturbation series calculated through third order essentially yield the J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules correct equilibrium internuclear distance and geometry, even in the case of H1 3 . For this molecule, the perturbation series is best converged in the vicinity of the equilibrium distance. For larger R, the truncation error remains small if one of the internuclear distances is held near the equilibrium distance. However, if all internuclear distances are increased, the truncation error increases as a consequence of the incorrect separated-atom limit of the zero-order wave function. In the case of HeH11 2 , the convergence of the perturbation series is significantly improved, particularly in the vicinity of the separated-atom limit. The present calculations can be improved without significant complication of the method by introducing a screened nuclear charge for the unperturbed Hamiltonian22 or by repositioning the unperturbed, single-center Hamiltonian away from the physical charges.3 Calculations incorporating these improvements and extending this initial study to higher-order are currently underway. For all diatomic and polyatomic systems examined here, the perturbation energy summed through third order generally yields results comparable in accuracy to variational calculations of moderate size and complexity. Given the simplicity and efficiency of the method, these results for simple systems are very encouraging. Since, through a given order in the perturbation, the computations are of a fixed degree of complexity regardless of the size of the molecule, application to larger systems with higher Z seems promising. 4929 1s s 2l s e m,l 1 ~ R a ! 52 e 1 ~ R a ! 1 e 1 ~ R a ! . 1,3 Here the e 21 L are the first-order corrections to the singlet and triplet 2S and 2 P states of the two-electron atom, while K U U L s e 2l 1 ~ R a ! 52 2l ~ r ! 1 2l ~ r! , ra s e 2s 1 ~ R !5 1 1 2 ~ 11 43 R1 14 R 2 1 18 R 3 ! e 2R , 4R 4R s e 2p 1 ~ R !5 3 1 1 R 3 4R 2 F G 3 1 ~ 71 114 R1 34 R 2 1 18 R 3 ! e 2R . 3 ~ 11R ! 1 R 4R The off-diagonal matrix 1Z C v 2s,2p (R C ), with V5Z B v 2s,2p (R B ) K U U L F G v 2s,2p ~ R ! 52 2s ~ r! 52 element, 1 2 p ~ r! ra 3 3 ~ 11R ! 21 4R 4R 2 1 81 ~ 31R1 41 R 2 ! e 2R . The complete first-order wave function is then given by APPENDIX: THREE-ELECTRON MOLECULES c 1 5c s c s1 1c p c 1p 1 gc 80 , where The zero-order wave function for the three-electron, three-center molecule is degenerate, while a more complex, permutational symmetry not present in the ground state of the corresponding two-electron molecule also appears here. Expressions for the first-order wave function and secondorder energy of this system are obtained here to illustrate how these affect the multiperturbation expansion. The doubly-degenerate zero-order wave function for this system is c 0 5c s c s0 1c p c 0p , where c l0 52 2 ~ 1/2 ! ~ 12 P 13 ! 1s ~ r1 ! 1s ~ r2 ! 2l ~ r3 ! for l5s or p. The coefficients, c l , are the eigenvectors of the perturbation matrix formed with these degenerate wave functions. The eigenvalues are just the first-order energies, e 15 e s1 1 e 1p e s1 2 e 1p 6 2 2 A S 12 2V e s1 2 e 1p 2 , appears as an admixture of the other degenerate, zero-order wave function ~without, however, contributing to the secondorder energy!. Also, m,l m,l c l1 5 c a,l 1 1Z B c 1 ~ R B ! 1Z C c 1 ~ R C ! , are the first-order wave functions for the where the ca,l 1 ground and lowest 2P-state of three-electron atoms. These can be expressed completely in terms of the first-order wave functions of two-electron atoms,23 1 2 ~ 1/2 ! c a,l ~ 12 P 13 ! $ c 11 S ~ r1 ,r2 ! 2l ~ r3 ! 12 2 ~ 1/2 ! 1 52 1 3 3 @ c 21 L ~ r2 ,r3 ! 1s ~ r1 ! 1 c 21 L ~ r2 ,r3 ! 1s ~ r1 ! 3 1 c 21 L ~ r1 ,r3 ! 1s ~ r2 !# % . The c m,l 1 (R a ) can be constructed entirely from first-order, H1 2 -like wave functions, s 2 ~ 1/2 ! c m,l ~ 12 P 13 [email protected] c 1s 1 ~ R a ! 52 1 ~ R a ,r1 ! 1s ~ r2 ! 2l ~ r3 ! where m,l m,l e l1 5 e a,l 1 1Z B e 1 ~ R B ! 1Z C e 1 ~ R C ! , with 1 1 S e a,l 1 5e1 1 and D c 08 52c p c s0 1c s c 0p 1 21L 3 23L e 1 e1 , 2 1 2 s 1 c 1s 1 ~ R a ,r2 ! 1s ~ r1 ! 2l ~ r3 ! s 1 c 2l 1 ~ R a ,r3 ! 1s ~ r1 ! 1s ~ r2 !# , s are solutions to the degenerate, first-order, perwhere c2l 1 turbation differential equation for the 2l zero-order wave function,10 e.g., J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 4930 Galvan, Abu-Jafar, and Sanders: Potential energy surfaces for polyatomic molecules s G 0 c 2s 1 ~ R a ,r ! 2 S D 1 This first-order wave function then yields the second-order energy coefficient, 1 ~ s, p ! s e a,2 ~ R a ! 5 21 @ e 21,1S,2p s ~ R a ! 1 e 21,1P,2s s ~ R a !# 1,1 1 s 1 e 2s 2s ~ r! 2 v 2s,2p ~ R a ! 2p ~ r! 50. 1 ra 3 3 1 23 @ e 21,1S,2p s ~ R a ! 1 e 21,1P,2s s ~ R a !# , and 1s 2 e 2 5 e 2 1c 2s e s2 1c 2p e 2p 1c s c p e s,p 2 , s ,2 p s e 2s ~ R B ,R C ,Q ! 522 1,1 2 where e 1s 2 is given by Eq. ~18!, the second-order energy for the ground state of an H1 3 -like molecule, 2 2l s 2 2l s a,2l s e l2 5 e a,l 2 1Z B e 2 ~ R B ! 1Z C e 2 ~ R C ! 1Z B e 1,1 ~ R B ! K FK U U L U U LG s c 2s 1 ~ R B ,r ! s 1 c 2p 1 ~ R B ,r ! 1 2p ~ r! rC 1 2s ~ r! rC 1,3 L,2l s 2 s One approach to evaluating the e2l 2 , e 1,1 coefficients can be found in Ref. 10. 2l s ,2l s s 1Z C e a,2l ~ R B ,R C ,Q ! , 1,1 ~ R C ! 1Z B Z C e 1,1 . s , and e2(s,p) 2 and 2 2 ~ s,p ! s e s,p ~ R B ! 1Z 2C e 22 ~ s,p ! s ~ R C ! 2 5Z B e 2 ~ s,p ! s ~ s,p ! s 1Z B e a,2 ~ R B ! 1Z C e a,2 ~RC! 1,1 1,1 s ,2p s 1Z B Z C e 2s ~ R B ,R C ,Q ! . 1,1 2 The multiperturbation coefficients appearing in this second-order energy coefficient are given by 1 3 1 2 L 3 2 L l e a,l 2 5 2e 2 1 2e 2 1 d 2 , where d l2 is the three-electron contribution to the secondorder energy of a three-electron atom,23 K 2l s s e 2l 2 ~ R a ! 52 c 1 ~ R a ,r ! U U L 1 2l ~ r! , ra 1 3 1 2 L,2l s s e a,2l ~ R a ! 1 23 e 21,1L,2l s ~ R a ! 1,1 ~ R a ! 5 2 e 1,1 with 1,3 L,2l s e 21,1 K ~ R a ! 522 1/2 c 21 1 U 1,3 L ~ r1 ,r2 ! U 1 r 1a L 1 1s ~ r1 ! 2l ~ r2 ! 62l ~ r1 ! 1s ~ r2 ! , r 2a K s ,2l s s e 2l ~ R B ,R C ,Q ! 522 c 2l 1,1 1 ~ R B ,r ! K s e 22 ~ s,p ! s ~ R a ! 522 c 2s 1 ~ R a ,r ! J. Goodisman, J. Chem. Phys. 47, 1256 ~1967!. R. L. Matcha and W. Byers Brown, J. Chem. Phys. 48, 74 ~1968!. 3 Z. Dvorácek and Z. Horák, J. Chem. Phys. 47, 1211 ~1967!; see also Z. J. Horák and J. Sisková, ibid. 59, 4884 ~1973!. 4 ~a! C. D. H. Chisholm, Mol. Phys. 21, 769 ~1971!; ~b! C. D. H. Chisholm and K. B. Lodge, ibid. 21, 775 ~1971!. 5 H. E. Montgomery, B. L. Bruner, and R. E Knight, J. Chem. Phys. 56, 1449 ~1972!. 6 F. C. Sanders, Phys. Rev. A 7, 1870 ~1973!. 7 Units of length and energy a 0 /Z and 2RhcZ 2 , respectively. 8 J. Hirschfelder, W. Byers-Brown, and S. T. Epstein, Adv. Quantum Chem. 1, 255 ~1964!. 9 B. Joulakian, Ind. J. Phys. 64B, 189 ~1990!. 10 C. D. H. Chisholm and K. B. Lodge, Chem. Phys. Lett. 15, 571 ~1972!; Mol. Phys. 28, 249 ~1974!. 11 C. W. Scherr and R. E. Knight, Rev. Mod. Phys. 35, 1033 ~1963!. 12 A. Dalgarno and N. Lynn, Proc. Phys. Soc. London 70, 223 ~1957!. 13 D. M. Chipman and J. O. Hirschfelder, J. Chem. Phys. 59, 2838 ~1973!. 14 I. N. Levine, J. Chem. Phys. 41, 2044 ~1964!. 15 W. Kolos, K. Szalewicz, and H. J. Monkhorst, J. Chem. Phys. 84, 3278 ~1986!. 16 W. Kolos and C. C. Roothan, Rev. Mod. Phys. 32, 219 ~1960!. 17 W. Kolos and J. M. Peek, Chem. Phys. 12, 381 ~1976!. 18 S. Peyerimhoff, J. Chem. Phys. 43, 998 ~1965!. 19 D. Frye, A. Preiskorn, G. C. Lie, and E. Clementi, J. Chem. Phys. 92, 4948 ~1990!. 20 D. Talbi and R. P. Saxon, J. Chem. Phys. 89, 2235 ~1988!. 21 D. F. Zetik, dissertation, The University of Texas at Austin, 1968; D. F. Zetik and R. D. Poshusta, J. Chem. Phys. 52, 4920 ~1970!. 22 C. D. H. Chisholm and K. B. Lodge, Mol. Phys. 22, 673 ~1971!. 23 Y. Y. Yung, F. C. Sanders, and R. E. Knight, Phys. Rev. A 48, 74 ~1972!. 1 U U L 1 2l ~ r! , rC U U L 1 2p ~ r! , ra J. Chem. Phys., Vol. 102, No. 12, 22 March 1995 Downloaded 20 Aug 2013 to 212.14.233.38. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

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