The Journal of Physical Chemistry, Vol. 83,
Diatomic Potential Energy Curves
No. 9, 1979 1221
Calculations of Potential Energy Curves for the Ground States of NaH' and KH' and States of NaH and KHla
n
Carl F. Melius,lb Robert W. Numrich,lc and Donald G. Truhlar*lc Sandia Laboratories, Livermore, California 94550, and Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received November 30, 1978)
Calculations of diatomic potential energy curves calculated by using two different kinds of effective core potentials are presented. The curves studied are the ground states of NaH' and KH' and the lowest 311and lII states of NaH and KH. The two methods yield qualitatively similar but quantitatively different results. The best values calculated for the potential well depths and positions for NaH+ and KH' are 0.06 and 0.02 eV and 5 . 1 ~ ~ and 6.5ao, respectively. Both methods predict that the potential curves of the Il states of the neutrals are qualitatively similar to the potential curves of the ground states of the ions.
I. Introducition Although NaH+ and KH+ are simple diatomic systems, their potential energy curves are not well studied. However, these potential energy curves are needed to understand ion-atom collisions2and to serve as input for diatomics-in-molecules studies of the potential energy surfaces for Na Ha and K H2 collision^.^ In addition, these systems are interesting for the theory of chemical bonding since each involves a single valence electron. In this article, these potential energy curves are calculated using effective core potentials to simplify the electronic structure problem. Two different kinds of effective core potentials4 are applied, and the results are compared. For the ground s8tateof NaH+ a potential curve has been calculated previously by Claxton and M~Williams.~ Single points on the potential curves of NaH+ and KH+ have been calculated by Gaspar and Tamassy-Lentei.' Most recently Valance7 has calculated well depths and distances for the ground states of NaH+ and KH+ and whole potential curves for some of their excited states. These previous results are compared with the present ones. Recently, Chu8 has pointed out that II state potential curves for un-ionized diatomics may give good approximations for the ground state potential curves of the corresponding ionized species. As a test of this conclusion, the ground-state ion curves reported here are compared with corresponding II state curves, one set reported here for the first time and the other set reported in a previous p u b l i ~ a t i o n .The ~ new calculations for the IT states also illustrate the use of a one-electron model for systems with two valence electrons.
+
+
11. Calculations for Ions A. Methods In and l p . The first method of this paper , is described loy two of the authors in previous paper^.^,^^ It is used heire to yield results for the ions as consistent as possible with the previous resultsgfor the neutrals. The ions are treated as one-electron systems with the Hellman-type model potentials of Schwartzl1-l3as the effective core potentials for the alkali cores. The effective core potentials arid the basis sets used here are the same as those used for the neutrals. They are described in detail elsewhere9J4imd are summarized briefly in the Appendix. In the previous treatmentg of configuration mixing in the 'X+ states of NaH and KH, the calculations were improved by adding semiempirical polarization potentials to the ionic diagonal matrix elements to account for that part of the ion-induced-dipole effects missing in our variational treatment?JO Thus the potential 0022-3654/79/2083-1221$01 .OO/O
V&R) = - ( a / 2 ~ 4 ) u - e x ~ [ - ( R / r ) ~ l ) (1) was added to the ionic diagonal elements of the electronic Hamiltonian in the valence-bond basis before completing the calculation. In eq 1R is the internuclear distance, and for the diagonal ionic lZ+ matrix elements, 01 represented the sum of polarizabilities of H- and M'. The cutoff parameters rNa and rKwere adjusted in such a way as to obtain the correct dissociation energies of the A lZ+states of the molecules. Since all valence bond configurations for MN" are ionic, the same kinds of polarization potentials were added to the diagonal elements of the Hamiltonian matrix for the present calculations on the ions. In this case, for basis functions centered on H, a equals CYH(= 4.5~03) and for basis functions centered on M, a equals aM+. The values used for o(M+ and o ~ Mare the same as bef~re.~,lO Notice that polarization of the valence electron of the alkali is obtained through the variational calculation, and so it is not included in the semiempirical polarization potential. In the calculations for the lZ states of MH the polarization potentials had large effects because of the large polarizability of H-. The polarizabilities of M+ and H are much smaller so the polarization potentials have much less effect on the present results. The calculations of the ionic potential curves as described above without and with the polarization potential are designated methods I n and lp, respectively. B. Method 2. The second method is a variant of the ab initio effective core potential approach of the first author and c o - w o r k e r ~ . ~ ~This - ~ ~ method involves "coreless" Hartree-Fock orbitals and matrix element fitting procedures. Unlike method 1, the effective core potentials of method 2 contain angular momentum projection operators. These potentials may be written jjwe
=
ucore L,,(r) -t
Lx-1
1
1=0
m=-1
C C Ilm)[UYre(r)- Urz(r)l(lml
(2) where r is the distance from the nucleus and Ilm) is a spherical harmonic centered at the same nucleus. For Na and K, L,, is taken as 2. The details of the effective core potential are given in the Appendix. Using these potentials, NaH+ and KH' are treated as one-electron systems using extended Gaussian basis sets described in the Appendix. These calculations of the ionic potential curves are designated method 2.
111. Calculations for II States of Neutrals A. Method 1. These calculations were reported previouslyg (although not in tabular form) and are included 0 1979 American Chemical Society
1222
The Journal of Physical Chemistry, Vol. 83,No. 9, 1979
TABLE I : -
-_
R,a , 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 11.0 12.0 13.0
C.F. Melius, R. W. Numrich, and D. G. Truhlar
Potential Energy Curves (hartree) for Ground States (X * E + )of Ionsa,b method I n
NaH’ method l p c
0.20023
0.19990
0.03526 0.01368 0.00470 0.00115 -0.00010
0.03513 0.01364 0.00467 0.001 09 - 0.00021
- 0.00044
- 0.00063
- 0.00024
- 0.00053
- 0.00011
-
- 0.00004
- 0.00042
0.00047
- 0.00002
- 0.00032
- 0.00001
- 0.00022
- 0.00000 - 0.00000
- 0.00014 -
0.00009
KH’ method l p c
method 2c
method In
0.12967 0.05 59 1 0.0217 0 0.00657 0.00034 - 0.00182 - 0.00223 - 0.00199 - 0.00160 - 0.00124 - 0.00095 - 0.00073 - 0.00057 - 0.00045 - 0.00033 - 0.00028 -0.00023 -0.00016 - 0.00011
0.33288
0.33277
0.07282
0.07274
0.01391 0.00562 0.00205 0.00058 0.00004
0.01388 0.00558 0.00198 0.00048 - 0.00009
- 0.00015
- 0.00033
- 0.00009
- 0.00031
- 0.00004
- 0.00028
- 0.00002 - 0.00001
- 0.00025 - 0.00019
- 0.00000 - 0.00000
-0.00013 - 0.00009
- 0.00008
a Zero of energy corresponds to M+ t H infinitely separated. methods.
here for comparison. We recall here that these are single-configuration valence bond calculations where the configuration is formed from basis function x4on the alkali and x1 on the hydrogen (see Appendix). These calculations do not involve the polarization potential since the configuration is neutral and covalent. They are designated method 1 for the II states. B. Methods 2H, 2 X , and 21. Method 2 is reported in the literature for the first time here.l* It can be applied to hydrides in which the partial ionic character involves a net displacement of negative charge toward the hydrogen nucleus. Many low-lying states of the alkali hydrides, including the II states of interest here, satisfy this cond i t i ~ n . ~InJ a~ generalized valence bond description of H-, its two electrons occupy different orbitals with different radial extents,20 with the inner orbital being very hydrogenic. In a generalized valence bond study of alkali hydrides it has been found21that one of the two valence orbitals is essentially the same as an unperturbed inner H- orbital. The other valence orbital may be determined variationally in the presence of the alkali core and the inner H--like orbital. In the present methods this inner H--like orbital is considered frozen; it may be considered to be a core orbital, and the interaction of the other valence electron with it is represented by an effective core potential. The determination of the effective core potential to represent the inner orbital of H- is discussed in an unpublished part of a thesisaZ2The effective core potential must be different for singlet and triplet two-electron states. Since the core contains only an s electron, it was considered sufficient to set L , equal to 1. Since H- has a bound state only for lS symmetry, the effective core potential could be obtained from ab initio calculations on H- only for this symmetry. However one of the advantages of the coreless Hartree-Fock method with matrix element fitting as described in ref 15-17 is that it can be used to obtain effective core potentials even for the 3S, IP, and 3P cases where the core does not bind an additional electron. Thus effective core potentials with the form of eq 2 were obtained with L,, = 1 for both the singlet and triplet.22 The effective core potentials are given in the Appendix. After the effective core potentials are obtained, the calculation of the potential curve of the neutral state of
method 2c 0.1311 5 0.0657 1 0.03164 0.015 1 2 0.00685 0.00261 0.00053 - 0.00040 - 0.00074 - 0.00080 -0.00073 - 0.00063 - 0.00052 - 0.00042 - 0.00034 - 0.00028 - 0.00023 - 0.00016 -0.00011 - 0.00008
1 hartree = 27.2116 eV = 4.35981 x
- aH/(2R4)
- 0.00549 - 0.00360 - 0.00246
-0.00174 -0.00126 - 0.00094 - 0.00071 - 0.00055 - 0.00043 - 0.00034 - 0.00028 - 0.00022 - 0.00015 - 0.00011 - 0.00008
J.
Preferred
MH involves two steps: (i) a one-electron calculation to determine the energy of the MH+ moiety; (ii) a second one-electron calculation in which the valence electron interacts with the M+ moiety through the same effective core potential as used in section 1I.B and with the H moiety through an effective core potential as already discussed in this section. The “core” for step ii is MH+ (as opposed to MH2+for the calculation by method 1) so that the total energy of the neutral state of MH is the sum of the total energy of step i and the one-electron energy of step ii. The basis sets for step ii are the same as used for the calculations of section 1I.B for the ions. Step i requires more consideration. One possibility is to use the ionic calculation of section 1I.B. This will be designated method 21; but the MH+ ion dissociates to’M+ H, and its potential curve is dominated by the ion-induced-dipole potential at large distances. However the neutral state of interest dissociates to M + H and the H atom does not interact with an ion at large distances. To eliminate this spurious interaction we repeated the calculation of section I1 including only a single contracted basis function on H and no basis functions on M+. This minimum basis set calculation yields the interaction energy of an unperturbed H atom with the M+ core. Use of this calculation for step i will be designated method 2H. To test the sensitivity of the calculation to the forzen-core assumption for H- we repeated the calculation for MH+ using all the basis functions of the original calculation except for p functions on H. This eliminates the long-range polarization effect. Use of this calculation for step i will be designated method 2x.
+
IV. Results and Discussion
The results for the ionic potential curves are in Table I, and the results for the II-state potential curves of the neutrals are in Tables I1 and 111. For these tables the zero of energy is for R = m in each case. 1. Ionic States. In order to compare the present results for ionic potential curves to the previous ones the results are reexpressed with the zero of energy for each calculation equal to the calculated energy for the ground state neutral atoms. The comparison for NaH+ with this zero of energy is shown in Figure 1. For the present one-electron calculations, the basis sets for both methods l and 2 include
The Journal of Physical Chemistry, Vol. 83,No. 9, 1979
Diatomic Potential Energy Curves
1223
TABLE 11: Potential Energy Curves (hartree) for 17 States of NaHa method lb
R, (2" 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 11.0
3n
method 2Hb
0.26616
0.27827
0.04131
0.04965
0.00428
0.0.0933
- 0.00056
0.00221
-0.00068
0.00073
-0.00037
0.00030
- 0.00018
0.00013
- 0.00008
0.00006
-0.00003
0.00003
method 21c
method 2X
r1
'n
'11
0.321 89 0.13053 0.047 69 0.0167 3 0.00485 0.00062 - 0.00069 -0.00093 - 0.00083 - 0.00065 - 0.00047 -0.00034 - 0.00023 -0.00017 - 0.00011 - 0.00008 - 0.00005 - 0.00003
0.33652 0.13762 0.05742 0.02433 0.01078 0.00510 0.00265 0.001'51 0.00093 0.00061 0.00041 0.00027 0.00020 0.00013 0.00009 0.00006 0.00004 0.00002
0.11969 0.05531 0.01856 0.00432 - 0.00126 - 0.00280 -- 0.00273 - 0.00219 - 0.00166 - 0.001 1 7 - 0.00083 - 0.00058 - 0.00041 - 0.00024 - 0.00020 -0.00015 -0.00010 - 0.00006
I11
'n 0.13432 0.06240 0.02828 0.01192 0.00467 0.00168 0.00061 0.00025 0.00011 0.00009 0.00005 0.00003 0.00002 0.00006 0.00000
3n 0.11950 0.04773 0.01528 0.00162 -0.00341
_-
'n
0.13412 0.05982 0.02501 0.00930 0.00252
- 0.00001 - 0.00001 - 0.00001
This method is applied only for Preferred methods. a Zero of energy corresponds t o Na(>P)+ H infinitely separated. distances whlere the potential energy of MH' as calculated by method 2 is greater than the energy of M' + H. TABLE 111: Potential Energy Curves for method lb
R, a, 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 11.0
n States of
KHa
method 2Hb
method 2X
method 21'
'n
'n
In
'17
'n
3n
0.49734
0.50558
0.09363
0.09999
0.01501
0.01930
0.00151
0.00408 0.00110
-0.00031
0.00040
-0.00016
0.00019
-0.00006
0.00010
-0.00001
0.00007
0.46094 0.1937 1 0.08405 0.03773 0.017 7 4 0.00883 0.00463 0.00254 0.00145 0.00087 0.00054 0.00035 0.00023 0.00015 0.00011 - 0.00007 - 0.00005 t 0.00002
0.12605 0.064 3 0 0.03241 0.01561 0.0067 3 0.00241 0.00040 - 0.00044 -0.00073 - 0.00078 -0.00059 -0.00053 - 0.00041 - 0.00031 - 0.00023 - 0.00016 - 0.00012 -0.00006
0.13548 0.07230 0.03908 0.02098 0.01114 0.00591 0.00313 0.00166 0.00086 0.00042 0.00029 0.00012 0.00006 0.00002 0.00001 0.00000 0.00000 -0,00001
0.12464 0.06032 0.02725 0.01159 0.00404 0.00041 -0.00117
- O.OOO30
0.45151 0.18571 0.07738 0.03226 0.01333 0.005 33 0.00190 0.00044 - 0.00014 - 0.00033 -0.00034 - 0.00030 -0.00024 - 0.00018 -0,00013 - 0.00009 -0.00007 -0.00003
'n
In 0.13407 0.06832 0.03392 0.01706 0.00845 0.00391 0.00156
a Zero of energy corresponds t o K(*P) + H infintely separated. Preferred methods. This method is applied only for distances whlere the potential energy of MH' as calculated by method 2 is greater than than the energy of M +t H.
valence p functionar centered on the alkalis and expanded functions. In addition, the basis for method 2 also includes p functions on H. The calculations of Claxton and McWilliams5 are all-electron, spin-projected, unrestricted-Hartree-Fock calculations with an extended Gaussian basis set without valence p functions on the alkalis or on H and without expanded functions. They considered only the region 2.8-3.8~~.Over this region they found a purely repulsive potential curve which rises by 1.2 eV as distance decreases. Figure 1 shows that the potential curves of the present calculations are less steep in this region and that their calculated repulsive wall lies between the results of method 1 and those of method 2. However, the Hartree-Fock method underestimates the ionization potential of Na by 0.2 eV,16323and the error in the method of Claxton and McWilliams is probably about the same. Thus to adjust their curve in Figure 1 to the same large R asymptote as the others it must be raised 0.2 eV. This puts it in good agreement with the curve for method l p (six basis functions). Claxton and McWilliams concluded from their calculations that NaH+ is unbound. We too find no minimum in the potential curve for R I 3.8a0,but all three calculations reported here predict a shallow well in the
potential curve at larger distances. This well is due to the ion-induced-dipole interaction and so valence p functions on the alkali are required to treat it accurately. Also shown in Figure 1 is the single point obtained by Gaspar and Tamassy-Lentei6 for NaH+. These authors used effective core potentials of the same Hellmann form as used in method 1but with the parameters of Szasz and M ~ G i n n . They ~ ~ used a single Slater-type orbital to represent the valence electron and optimized the position of the center of this orbital variationally. The point shown in Figure 1is their best value and corresponds to an orbital centered 3 . 5 from ~ ~ the Na nucleus for R = 3 . 5 6 ~ Thus ~. the orbital is very close to being centered at the H nucleus. This result agrees with the present results since it is found here that the wave function for the ground state has mostly the character of an H (1s) orbital. The results for KH+ are similar to those for NaH+. For KH+ the calculation of Tamassy-Lentei at 4 . 2 4 lies ~ ~ 1.0 eV higher than the value interpolated from method lp. Next we consider the work of Bellomonte et These authors used a single basis function expressed in elliptic coordinates with two nonlinear variational parameters. They do not give any other details of their calculations for
1224
The Journal of Physical Chemistry, Vol. 83, No. 9, 7979
NOH
,,
,Cloxtan
i
t
'--i
t a0""
2:
" " I *
" I " '
15
" "
A 3
F. Melius, R. W. Numrich, and D. G. Truhlar
TABLE IV: Calculations of Binding Parameters for the Ions
+
ond McWilliorns
C.
"
!
45
R(o,l
Figure 1. Comparison of calculated potential curves for NaH'. In order to put all the curves on the same energy scale, the zero of energy for each calculation is the calculated energy for the ground state neutral alkali atom infinitely separated from ground state H as calculated by that method. I n addition to the full calculation for method Ip, which employed six contracted basis functions, we show the results obtained when the second and third Hcentered contracted functions are omitted.
the alkali hydride ions except that "We did not succeed in establishing the stability for this series of ions." This presumably means that the potential curves were purely repulsive, and this presumably is due to the inability of their basis functions to account for the polarization effect. Their discussion involves the masses of the nuclei, but these play no role in the Born-Oppenheimer adiabatic framework used by them and us. The repulsive walls calculated by the two methods studied here are in agreement; this is encouraging. In the repulsive region the semiempirical polarization potential is less important than a t larger distances. At larger distances method 2 predicts a deeper minimum and a smaller equilibrium internuclear distance than method In. These differences may be manifestations of the deficiencies of the model potential and basis set of method 1. Previously it was found that method 1 underestimated the dissociation energy and overestimated the internuclear distance for the NaH and KH ground state^.^ This was attributedg to the fact that the basis set is more optimum for separated atoms than for the neutral molecule. The semiempirical polarization potential improves the binding energy but not enough. Method l p leads to a more attractive potential curve than method 2 does at large R, although both include the long-range ion-induced-dipole polarization interaction. Method l p includes this effect through the semiempirical polarization potential whereas method 2 includes it variationally. Since the ion dissociates to M+ + H, the asymptotic form of the potential curve is -aH/(2R4). This quantity is tabulated in Table I for large R for comparison with the calculations. The calculated potential curves are more attractive than this for R L loao, but for R < 9ao method I p underestimates the attractive interaction. The values used for YM (see eq 1) in this work are the fairly large values ( 7 N a = 9.3ao and 7K = 9.9ao) obtained previously for the lZ+ states of the neutrals. These large values are clearly responsible for method l p underestimating the polarization interaction at medium r. Clearly, if YM were reoptimized for the ions, better agreement could be obtained between methods l p and 2. In method 2
KH'
NaH' calculation
Re, a,
De, eV
Re, a,
De,eV
Valance method I n method l p method 2
5.8 6.1
0.02 0.012
10 7.3
0.004
6.3 5.1
0.018 0.061
7.4
0.010
6.5
0.022
0.025
where the polarization effect is obtained variationally without semiempirical parameters, the asymptotic form of the potential curve is obtained within 10% at distances down to 6ao for NaH+ and 7 . 5 for ~ ~ KH+. Because of the larger basis set and the more complete inclusion of the polarization effect, method 2 is expected to be more accurate than method 1 for the ionic potential curves. Fitting parabolas to the minima of the potential curves obtained with method 2 for the ion yields the following best values for the well parameters: D,(NaH+) = 0.061 eV, R,(NaH+) = 5.07a0, D,(KH+) = 0.022 eV, R,(KH+) = 6 . 4 8 ~ .Calculations of these quantities have also been reported recently by Valancea7 His values are compared to our values as obtained by all three methods in Table IV. Valance used Hellmann-type effective potentials, as we did for method 1, but he used the parameters of Hart and Goodfriend.26 We prefer the parameters of Schwartz for reasons detailed e l ~ e w h e r e . ~ J ~ Valance used a basis of one s function and one p function on M and two s functions and one p function on H, as compared to two contracted s functions and one contracted p function on M and three contracted s functions on H in method 1. The large distance, 10ao,he obtains for the KH+ potential well is quite surprising. Comparison to method I n in Table I shows that little repulsive interaction is predicted at distances beyond 6 ~ 0 .Yet the contracted basis functions on K in method 1 are carefully chosen to have accurate long-range tail~.~JOSimilarly, method 2, which involves an extended basis set, predicts little deviation from the attractive polarization interaction a t distances beyond 7ao. If there is no repulsion at loao,the well cannot be that far out. Further Valance's well depth corresponds to a net attractive energy of 0.00092 hartree. Comparison to Table I shows that this considerably exceeds the value calculated from the asymptotic form of the polarization interaction. We conclude that Valance's calculation for KH+ is inaccurate due to deficiencies of the effective core potential, the basis, or both (or an error in his calculation). For NaH+ his calculation predicts too small a well depth, like the present method 1. 2. II States. Since no ionic configurations are included in method 1 for the II states of the neutrals there is now only one version of method 1, but there are three versions of method 2 corresponding to the three treatments of the MH+ moiety. Method 2H is the consistent version of method 2; it treats the inner orbital as frozen in both step i and step ii. Methods 2X and 21, the latter applicable only at small R, allow for some relaxation of this "core" orbital in step i but not step ii. It is encouraging that the consistent version, method 2H, is in best agreement with method 1 at both small R and large R. One might expect that the frozen core assumption is violated to some extent but method 2X probably overestimates the amount of relaxation of the inner orbital. This is because method 2X, like method 21, estimates the relaxation in a calculation on MX+, but the actual relaxation in the presence of a second electron in MX would be expected to be less than in MX+. Method 1 allows no relaxation of the orbital on H for the neutral II states because it is a single-configuration valence bond calculation with frozen atomic orbitals.
The Journal of Physical Chemistry, Vol. 83, No. 9, 1979
Diatomic Potential Energy Curves TABLE: V: Primitive Basis Sets method 1
A Na
-.
basis function 1 2 3 4 5, 8 6, 9
7, 10
K
H
2
Na
K
1 2 3 4 5, 8 6, 9 7, 1 0 1 2 3 4 5 1 2 3 4 5 6, 13 7. 1 4 8, 1 5 9, 1 6 10, 1 7 11,18 12,19 1 2 3 4 5, 11 6, 12 7, 1 3 8, 1 4 9, 15
10,16
H
1225
1 2 3 4 5 6
7 8 9 10 11 12 13 14 15, 1 6
pj
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2
nj
3 4 3 3 3 3 3 3 4 4 4 3 4 4 1 1 2
1 2
1 1 1 1 1 2
2 2
2
2 2 2 2 2 2 2 2 2 2
2
2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2
Thus the goo'd agreement of methods 1and 2H may be due a t least in part to both methods underestimating the relaxation of the inner orbital on H. Nevertheless, considering the differences in basis sets and approaches, the agreement of methods 1 and 2H is remarkably good. Method 2H predicts qualitatively the same singlet-triplet energy diffeirences as method 1 for both molecules at all R , and methods 21 and 2X predict exactly the same singlet-triplet separation as method 2H. This is especially noteworthy because methods 2H, 21, and 2X obtain these energy diffeirences solely as a consequence of using different effective core potentials for the singlet and the triplet in the one-electron calculations of step ii. In method 1 the singlet--triplet energy differences are a consequence of two-electron exchange integrals as usual. It is interesting to compare the ionic potential curves to those for ,the II states. In method 1 the ion is treated
2 2 3 3
1 1 1 1 2 2 2 2 3 3
1 1 1 1 1 1
1 1 1 2 2 2 3 2 2
fja
ai
bj
0.901 0.447 1.180 0.728 0.616 0.890 0.535 0.730 0.405 0.750 1.187 0.514 0.950 0.570 1.000 1.2048 1.2048 0.3178 0.3178 1.2198036 0.4570551 0.0634131 0.02642871 0.008 4.3305615 0.1308353 0.0464242 0.01 74370 0.008 0.0220458 0.008 0.43740 0.038 12 0.01485 0.008 0.13944 0.03863 0.01441 0.008 0.02485 0.008 68.16000 10.24650 2.34648 0.67332 0.22466 0.08221 7 0.0197 09 5
0.01 0.0054009 0.278588 0.0671 157
0.02 0.01 0.03 0.009
in terms of six one-electron basis functions and the II states of the neutrals are treated in terms of one two-electron configuration. In method 2H the ion is treated in terms of a large one-electron basis set and the II states of the neutral are treated by a one-electron calculation involving the same basis set. In both methods the shapes of the II state potential curves are in excellent agreement with the corresponding ionic curves, but for both systems the ionic curves are a little less repulsive than the average of the n state curves. For method 21 the relationship of the ionic and II-state curves is even more clear. For all distances for both Na and K, the ionic curve calculated by method 21 is intermediate in energy between the 311 and lII curves when all three are measured relative to the same asymptote. Thus the methods studied here confirm the general conclusion of Chu8 that these curves should be similar. To make more quantitative statements about the
1226
C. F. Melius, R. W. Numrich, and D. G. Truhlar
The Journal of Physical Chemistry, Vol. 83, NO. 9, 1979
TABLE VI: Contraction Coefficients method 1
A Na
CJi 0.192011@, t 0.012124@, t 0.3228370, t 0.5173180, X , = -0.211937@, t 1.10340@, - 0.149006@, - 0.16787904 x3+, = 0.184711@,+,a t 0.212063@,+,, t 0.636163@,,,, x, = 0.3737140, + 0.0374780, + 0.2770240, t 0.365014@, x 2 = - 0.191244@, t 1.137960, - 0.174774@, - 0.2362410, x,+, = 0.419879@,,,, t 0.3429270,+,, t 0.281232@,+,, XI = $1 x 2 = 0.78750, t 0.21283@, t 0.039240, - 0.00530@j x 1 = - 0.00203$, - 0.0088710, t 1.06221@, t 0.00255@, x , = 0.37876970, t 0.6603811@, X z + O = @?+a xS+,,a = - 0.0234999@,+,, T 1.0012794$~,,,
K H Na
2
D
0
x, =
XhiatD =
K
O~+-LY+R
0, 1 0, 1
0-2
0, 1 0, 1
0-4 0-3 0-5
Xi+D = 01+0 0, 1
= 0‘l.hck.P = 0.002550, t 0.01938@, t 0.0928@, t 0.294304
X4+a+D
xI
H
X21O
0-11
= 0s+D
TABLE VII: Parameters of Effective Core Potentials method 1 2
A Na K Na
Pka
“kol
S
1
S
1
-1 -1 -2 0 0 0 -2 -2
CY
S-D
P-D
2
0 0 0
D
K
2
S-D
2
P-D
2
D
2
-1 -1 -1 0 0 0 -2 -2 0 -2 0 0 -1
-1
-1
H
H
singlet S-P
2
singlet P
2
triplet S-P
2
triplet P
2
0 0’ 0 0 -1 -1 -1 -1 0 0 0 0 0 -1 -1 -1
0 0 0
relation of the ionic curve to the II-state states requires calculations on the latter in which the amount of relaxation of the orbital on H is optimized variationally.
V. Conclusions The effective core potentials described here seem to work well for ions and the present calculations of the potential curves for the ground states of NaH’ and KH’
rkol
2.267 1.866 0.935 28.4200 0.6105 0.07621 3.9264 1.7704 2.8630 1.368 0.3031
0.0 12.2245 1.083 4.2008 1.7972 0.2280 0.3538 0.08464 1.0910 0.4676 0.1496 0.4673
0.0 0.6990 0.2042 0.3325 0.8046 0.1629 0.0353
0.0 1.0194 0.5845 0.0532 1.0974 0.2923 0.5595 0.2644 0.1612
0.0 1.2700 0.1670 2.7238 0.2676 0.07875
Cka
14.0 18.0 5.8539 29.4418 1.5541 0.006673 51.5653 -46.9069 87.5679 16.1439 3.5816 10.0 1.1004 - 11.0951 9.3093 10.2369 - 0.02200 6.85 0.5086 - 0.7927 3.4940 0.2874 1.1498 18.0 - 5.3450 - 0.2924 -0.1675 0.4670 - 0.3446 -0.0508
1.0 - 0.09608
- 0.9430 0.01742 0.6689 0.3102 -1.0643 1.2660 - 0.4925 1.0 -0.9270 - 0.07300
1.0188 -0.2691 -0.004215
are the most reliable available potential curves for these cases. For the lII and 311 states of NaH and KH the oneelectron and two-electron calculations are in reasonable agreement with each other. The calculations are hard to compare in critical detail because method 1correctly treats both electrons but the calculations by method 2, being easier, were carried out with a more complete one-electron
The Journal of Physical Chemistry, Vol. 83, No. 9, 1979
Diatomic Potential Energy Curves
basis set. According to both methods though, the potential curves for the II states of the neutral are similar to the corresponding potential curves of the ions. The general success of the one-electron method for states of neutral MH is encouraging because electronically nonadiabatic collisions are easier to treat27in one-electron systems than in multielectron systems. Appendix For completeness, the details of all the basis sets and effective core potentials used in this work are given here. In general the basis functions are contracted functions of the form
xi(?) = CCjidj(7 - 7 ~ ~ ) 1
(AI)
where FA is the position of nucleus A, the primitive basis functions are &(?) = NJZaJxbJrnJ-aJ-bJ-l exp(-{fPJ)
(A21
and the z axis corresponds to the internuclear axis. The primitive basis sets are given in Table V, and the contracted sets are given in Table VI. The effective core potentials have the form of eq 2 with U y ( r ) = Ccknrnkaexp(-lk,rPkm) k
(A3)
where LY may be L,,, or " I - L,,". The latter is used to denote Ucore(r)- U f p ( n ) , L,,, is 0 for method 1 and is 1 for singlet or triplet H- in method 2 . It is 2 for alkalis in method 2 . The parameters of the effective core potentials used here are given in Table VII. The complete effective one-electron potential for a given center is the effective core potential plus the negative electronic-nuclear attraction term. The effective potentials for method 1 are explained el~ewhere.~Jll4 The basis sets for method 1are contracted Slater-type orbitals. The primitive bases consist of four s-type and three p-type functions on the alkalis and five s-type functions on H. These are contracted to two s-type and one p-type function on the alkalis and three s-type functions on H. A given primitive function sometimes occurs in more than one contracted function. Two of the s-type functions, x1 and xzon H, are very similar: one is the 1s eigenfunction of H and the other is a contracted function which provides a good representation of the inner orbital of H-. Both functions are included for consistency with ref 9 where dissociation to M+ H- was an important consideration. The effective potentials for method 2 were constructed using the matrix-element-fitting procedure described el~ewhere.l~-l'~ The H- effective potentials are described elsewhere.22 For the alkalis, instead of requiring the effective potential to reproduce the ab initio Hartree-Fock ionization potentials of the alkalis, they were adjusted to reproduce the experimental ionization potentials.28 Thus
+
1227
the effective potentials used here differ slightly from those given el~ewhere.'~-'~ Using these effective potentials four-term Gaussian s and p basis sets were optimized for Na and three-term Gaussian s and p basis sets were optimized for K. For H, a six-term Gaussian s basis set was taken from Huzinaga.29 These functions were augmented with expanded s basis functions to better describe Rydberg states and H- and with p basis functions to better described bonding. This led to the primitive basis sets for method 2 listed in Table V. A few of the tighter basis functions were contracted as described in Table VI. The whole basis set was used for calculations on the ion and for step ii of calculations on the neutral. For step i of calculations on the neutral, the whole basis was retained for method 21, but modifications described in section 1II.B were made for methods 2H and 2X. References a n d Notes (a) Thii work was supportedin part by the NatlonalScience Foundation and the U. S.Department of Energy. (b) Sandia Laboratories. (c) University of Minnesota. See, for example, A. Valance and G. Speiss, J . Chem. Phys., 63, 1487 (1975); R. E. Olson, E. J. Shlpsey, and J. C. Browne, Phys. Rev. A , 13, 180 (1976), H. Scheidt, G. Speiss, A. Valance, and P. Pradel. J . Phvs. 6 . 11. 2665 (1978): D. C. Haueisen. H. Mahr. J. C. Cassldy, C:L. Tang, D. A. Copeland, and P. L. Hartman, Bull. Am. Phys. Soc., 23, 19 (1978). See, for example, J. C. Tuliy, J. Chem. Phys., 59, 5122 (1973). A discussion of different kinds of effective core potentials is given in L. R. Kahn, P. Baybutt, and D. G. Truhlar, J . Chem. Phys., 65, 3826 (1976). T. A. Chxton and D. McWilliims, Trans. Farahy Soc., 66,513 (1970). R. Gaspar and I. Tamassy-Lentei, Acta Phys. Acad. Sci. Hung., 38, 3 (1975). A. Valance, Chem. Phys. Lett., 56, 289 (1978). S. Y. Chu, J . Chem. Phys., 64, 131 (1976). R. W. Numrich and D. G. Truhlar, J . Phys. Chem., 79, 2745 (1975). See also R. W. Numrich and D. G. Truhlar, J. Phys. Chem., 82, 168 (1978). W. H. E. Schwarz, Theor. Chim. Acta, 11, 377 (1968). W. H. E. Schwarz, Acta Phys. Acad. Sci. Hung., 27, 391 (1969). W. H. E. Schwarz, J . Chem. Phys., 54, 1842 (1971). R. W. Numrich, Ph.D. Thesis, University of Minnesota, Minneapolis, 1974. C. F. Melius, Ph.D. Thesis, California Instituteof Technology, Pasadena, 1973. S.Topiol, J. W. Moscowitz, and C. F. Melius, J. Chem. Phys., 68, 2364 (1978). C. F. Melius and W. A. Goddard, 111, Phys. Rev. A , 10, 1528 (1974). A similar one-electron model was used previously to calculate the large R avoided crossing in H, to study H+ 4-HH" -t H. A preliminary report of that work is C. F. Melius, Bull. Am. Phys. Soc., 19, 1199 (1974). C. F. Melius and W. A. Goddard, J . Chem. Phys., 56, 3348 (1972). W. A. Goddard, 111, Phys. Rev., 172, 7 (1968). C. F. Meiius and W. A. Wdard, 111, J. Chem. phys., 56,3348 (1972). See Figure 2. Reference 15, pp 116-120. E. Clementi, IBMJ. Res. Devel., Suppl., 9, 2 (1965). L. Szasz and G. McGinn, J. Chem. Phys., 42, 2363 (1965). L. Bellomonte, P. Cavaliere, and G. Ferrante, J . Chem. Phys., 61, 3225 (1974). G. A. Hart and P. L. Goodfriend, J. Chem. Phys., 53, 448 (1970). See, e.g., C. F. Melius and W. A. Goddard, 111, Phys. Rev. A , 10, 1541 (1974). Reference 15, p 62. S. Huzinaga, J . Chem. Phys., 42, 1293 (1965).
-
