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Semi-empirical calculation of the potential curves of NaNe, Na+Ne and Na−Ne
Volume 173, number 5,6
CHEMICAL PHYSICS LETTERS
19 October 1990
Semi-empirical calculation of the potential curves of NaNe, Na+Ne and Na-Ne * E. Czuchaj ‘, F. Rebentrost, H. Stall ’ and H. Preuss * Max-Planck-lnstitut fir Quantenoptik, Ludwig-Prandtl-Strasse
10.~8046
Garching near Munich, Federal Republic
ofGermany
Received 20 June 1990;in final form 2 August 1990
The ground-state potentials for Na+Ne, Na-Ne and NaNe as well as the A *Hand B ‘E potentials for NaNe have been calculated in the semi-empirical approach. The semi-empirical /dependent pseudopotential has been applied to simulate the effect of sodium core electrons, whereas the electrons of the neon atom have been kept frozen in the ground-state configuration. Polarization terms have been included in the interaction Hamiltonian to account for correlation effects. The SCF and valence CI calculations have been carried out in a Gaussian basis set. The calculated potentials are in good agreement with the experimental data and other calculations.
I. Introduction The results of our recent pseudopotential calculations on alkali-neon [ 1] and calcium-helium (neon) [ 2 ] systems exhibit an excessively repulsive character of all analyzed Z potential curves. It seems to us, one of the presumable reasons for this might be that the rare-gas atom pseudopotentials are still questionable. In this paper we present the results of a somewhat modified version of the pseudopotential approach. The interaction problem has been treated here in a non-symmetric manner, i.e. the sodium atom is considered as a one-electron system, because only its valence electron is treated explicitly. The effect of the core electrons is simulated by the corresponding semi-empirical I-dependent pseudopotential. Contrary to this, the neon electrons are kept frozen in the atomic ground-state configuration (the zeroth-order Hartree-Fock determinant). Due to this, the Coulomb and exchange interaction between the two atoms can be calculated directly. However, * Work supported partially by the Polish Ministry of National Education project CPBP 0 I .06. ’ Institute of Theoretical Physics and Astrophysics, University of Gdansk, Wita Stwosza 57,80-952 Gdadsk, Poland. ’ Institut fur Theoretische Chemie, Universitiit Stuttgart, Pfaffenwaldring 55, D-7000 Stuttgart 80, Federal Republic of Germany.
to allow for correlation effects in an efficient and simple way, we have incorporated the polarization terms in the interaction Hamiltonian. The use of the I-dependent pseudopotentials in molecufar calculations is described in detail in our previous papers [ 1,21, so it will not be repeated here. The present method is sketched in section 2. Details of the calculations are given in section 3 and the results obtained are discussed in section 4.
2. Method The calculation of the adiabatic potentials El(R) between the alkali atom (A) and rare-gas (RG) atom (B) in the Born-Oppenheimer approximation reduces to finding a solution to the Schriidinger equation for any internuclear distance R (in atomic units)
C&(R)
fl(I*,
rB, R) 9
(1)
where HA and Hz, are the Hamiltonians of the isolated atoms A and B, VABstands for the interaction between the two atoms, r,., and rz are the position vectors of the electron with respect to the alkali core and the RG atom, respectively, and R is the position vector of B with respect to A. In the present ap-
0009-2614/90/$ 03.50 Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
513
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preach, the Hamiltonian HA is the same as the corresponding one in ref. [ 11. On the other hand, we now take the Hamiltonian Ha as a ten-electron atomic Hamiltonian containing additionally the polarization terms. The latter are expressed as
v,(r,)= -
2B w4(rB,4 -
aq-6P, 7
B
W%B,6) ,
where lydand (Y,are the static dipole and quadrupole polarizabilities of the RG atom, respectively, and p, is the dynamical correction to (Y,. The cutoff function w is taken in the Gaussian form (3)
with the cutoff parameter 6. The interaction Hamiltonian VA, is defined as vABhj
rB,
R)
=V,,(r*,rB,R)+Vec(rA,rB,R)+Vcc(R),
(4)
where V,,describes the “pure” electron-electron and electron-nucleus interaction, I’,, involves all the socalled cross terms or three-body interaction terms and V,, represents the core-core interaction. All polarization terms include the cutoff functions defined as appropriate powers of eq. (3). More details on the one- and two-valence electron Hamiltonians used in the present approach are to be found in refs. [ 1,2]. Finally, the core-core interaction is expressed as 1/,(R) = K,(R)
-
Z:%
2~4
w4(R,6) -
Gff, 2R6
w6(R,4
where (Y,and S, stand for the dipole polarizability and the cutoff parameter of the alkali core, respectively, Z,, is the net charge of the alkali core seen by e- at an infinite distance (Z,= I ), and Z, is the RG atom nuclear charge (Z,=lO). In contrast to our previous calculations the short-range term I’,, encompassing the Coulomb interaction and exchange effects need not to be calculated here separately. It has been calculated directly within the SCF procedure. A detailed description of including an effective core-polarization potential in all-electron SCF and 574
valence CI calculations is given by Mueller and Meyer [ 31. In contrast with us, however, they did not use pseudopotentials and frozen atomic orbitals.
3. Calculations
(2)
w(r, 6) = 1-exp( -6r2)
19 October 1990
All the calculations have been performed with the MELD program [ 3 1. First, the SCF atomic orbitals for the ground-state configuration of the neon atom were obtained in the ( 5~3~) Gaussian basis set taken from Dunning [ 5 1. The Hartree-Fock energy for neon (polarization terms included in the interaction Hamiltonian) amounts to - 128.542587 au, which differs only slightly from the corresponding energy - 128.540448 au obtained by Dunning. Construction of the Gaussian basis set for Na, Na+ and Nawas based on the basis set reported by Mueller et al. [3] for the calcium atom. By dropping their eight largest exponents and adding two diffuse (0.0032 and 0.001) ones for s symmetry and, respectively, dropping six and adding two (0.0052 and 0.0021) exponents for p symmetry, we constructed the basis set for the negative ion of sodium. Finally, we took the (lOs9p3d/8s7p2d) basis set for Na- with the contraction coefficients -0.000474, 0.002696 and 0.007479 for s symmetry, 0.000045, 0.000201 and 0.000439 for p symmetry and 0.047965 and - 0.147777 for d symmetry. The calculated electron afinity for the (3s’) ‘S state of Na- amounts to 0.5453 eV, which agrees very well with the experimental value 0.546(5) eV [6]. For the (~P’)~P” state we have found the electron affinity to be 0.063 eV, which agrees with the theoretical value given by Norcross [ 7 1. Some of the exponents of the Na- basis set proved, however, insignificant in reproducing the experimental energies of a few lowest terms of the sodium atom. In consequence, in constructing the basis set for Na and Na+, we dropped the exponents 10.984, 0.2 and 0.00 1 for s symmetry and 1.763,0.0052 and 0.0021 for p symmetry. In addition, we added the exponent 0.008 for p symmetry. Finally, we used the (7s7p3d/%Sp2d) basis set with the contraction coefftcients 0.002189,0.014735 and -0.203912 for s symmetry, - 0.00399 1, -0.027910 and 0.082595 for p symmetry and 0.035585 and 0.078725 for d symmetry. The contraction involves the largest ex-
CHEMICAL PHYSICS LETTERS
Volume 173, number $6
ponents and was done to reduce the primitive Gaussian basis with slight degradation in atomic energies ( < 10m4au). The calculated ionization energies for a few lowest terms of the sodium atom along with the corresponding experimental values are compiled in table 1. The atomic basis set is, of course, too small to yield good agreement for higher terms. The size of the basis set for Na is motivated by the actual moIecular calculations. Namely, we wished rather to check whether the method works at all, and we were not interested in extensive molecular calculations. The only parameter needed to be determined in the present calculations is the cutoff one, 6, for neon (cf. eq. (2)). All the other parameters required in the calculations are taken from ref. [ 11. It has been found that the well-depth of the ground-state potential curve for NaNe depends very slightly on the cutoff parameter occurring in eq. (2). In contrast, the well depth of the A211 potential depends on 6 very strongly. Therefore, we decided to determine the value of 6 by matching the well depth of the A’II potential for NaNe to its experimental value determined by Havey et al. [ 91 from the temperature dependence of the intensity in the far-red wing of the Na resonance line perturbed by Ne gas. The value of 6 found in this way is 0.115 au. As can be seen in fig. 1, this value of 6 results in quite a realistic shape of the long-range interaction between a free electron and the neon atom. This is in drastic discrepancy with the long-range part of the pseudopotentials for rare gases used in our previous calculations [ 11, where the polarization terms are damped only at very short distances from the RG atom nucleus. The total pseudopotential (with its short-range term ) behaved, however, correctly. The same value of 6 has been used in the calculations on Na+Ne and Na-Ne. Table 1 Sodium-atom ionization energies (au) Term
(3s) ‘S (3P) *p (4s) ?s (4P) *p
Energy talc.
exp.
0.18881 0.11155 0.07155 0.05057
0.18886 0.11156 0.07158 0.05094
[81
19 October1990
Fig. 1. Long-range interaction between an electron and the neon atom defined by eq. (2) with the cutoff parameter 6= 0.1 I5 au.
As is well known, the so-called “basis-set superposition error” (BSSE) causes serious troubles in calculating interatomic interaction energies, especially if the basis sets are small. This problem has appeared in the present calculations as well. Our preliminary calculations on NaNe yielded the welldepth for the X ‘C potential to be much larger than its experimental value ( x 8 cm-‘). It has been found that the BSSE is responsible for this discrepancy. In the present approach (atomic orbitals of neon kept frozen), the molecular calculations were carried out in the sodium basis set augmented by the Is, 2s, 2p, 2p, and 2p, atomic orbitals of neon. In consequence, the SCF energy of the neon atom calculated in the molecular basis set weakly depends on the internuclear distance R. This insignificant effect for neon appears to be of great importance for the ground-state potential of NaNe. Representative values for the BSSE for NaNe are - 50, - 20 and - 5 cm-’ for internuclear distances of 5, 10, and 15 au, respectively, as obtained by subtracting the SCF energy of neon calculated in the atomic basis set from the corresponding SCF energy calculated in the molecular basis set in the interesting range of the internuclear separation [ lo]. Although the BSSE is quite negligible in the repulsive region of the X2C potential, it influences the attractive part of the potential very strongly and must be taken into account in the molecular calculations. 4. ResuIts and discnssion The reported molecular calculations have been 515
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CHEMICAL PHYSICS LETTERS
performed for the NaNe, Na+Ne and Na-Ne complexes in the internuclear separation range between 1.5 and 40.0 au. Since the basis set used for sodium is insufficient to describe the Na terms higher than 3p with satisfactory accuracy, we restrict our discussion to the lowest X 2Z, A 211and B 2C potential curves. Numerical values of the calculated potentials are available upon request from one of the authors (EC). The calculated ground-state potential for NaNe is shown in fig. 2. The well depth of the X 2 potential (De=9 cm-‘) differs from that of our previous calculations [ 1] (D,=4.5 cm-‘) but agrees cm-‘) with the experimental value (D,=8.1{9) [ 111: The location ‘ofthe potential minimum on the R axis is, as in the case of our previous calculations (R,= 11.4 au) shifted from its experimental value (R,= 10.0 au) to the right by about 1 au. Our A ‘ll potential has been determined by matching the well depth of the calculated potential to its experimental counterpart. As seen in fig. 3, the potential minimum (De= 165 cm-‘, R,=5.3 au) is in good agreement with th’e experimental findings, although its location on the R axis seems to be slightly shifted from
19October 1990
-
I
1
I
300Na Ne
200 -
B2zz
A%
loo-
-i Si o‘;
-100-
I
5
.,
,
,
I,
IO
/
,
L
I,
I5 R (a.4
Fig. 3. The fmt excited potential curves for NaNe: A, Experimental value found by Havey et al. [ 91; 0, experimental value determined by Gable and Winn [ 121; 0, experimental value from refs. [ 11,131; 0, theoretical value obtained by Peach [ 141; I, calculated by Philippe et al, [ 151.
the experimental value to the right. In turn, the B 2 potential with its very shallow minimum (D,=3.0 cm-l) located at R,=15.3 au agrees with the experimentaldata (D,=3.0(5) cm-‘, R,=14.4(3) au) [ 111. In general, one has to state that the present potentials for NaNe are in better agreement with the experimental data than that of our previous pseudopotential calculations [ 11. It seems, however, that they are still too repulsive and the reason for this is as yet unknown. The SCF calculations carried out for the groundstate interaction of the Na+Ne system have yielded the potential curve depicted in fig. 4. The results of the SCF calculations supplemented by the dispersion term ( - C6/R6) yield the potential minimum, which lies near the theoretical minimum recently found by Ahlrichs et al. [ 161. Note that their calculations in-
Ok!++==I
5
I
I
I
I
I
10
I
I
I
l5 R(a.u.1
Fig. 2. Ground-state potential curves for NaNe and Na-Ne. A, Potential minimum for NaNe determined by Lapatovichet al.
1111. 576
19 octoba 1990
CHEMICAL PHYSICS LETTERS
Volume 173, number 5,6
0.4-
oc -
‘; ti 0.3.5 >
-2OO-
k ;:
; 5 0.2s
-4oo-
-600I
a
0.1-
/
5
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15
Fig. 4. Ground-state potential curve for Na+Ne: - - - present SCF calculations; present SCF calculations plus -C6/R6 dispersion term taken from ref. [ 161; A, potential minimum found by Ahlrichs et al. [ 161.
elude the induction and dispersion terms of orders higher than Rm6.Finally, the SCF/CI pseudopotential calculations have been performed to obtain the ground-state potential curve for the Na-Ne system. In this approach, only two electrons of Na- were treated as the active ones. For each value of R, the SCF calculations were performed on the ground state of the system to obtain spatial orbitals. The latter were used to construct a basis set consisting of OTthonormal configuration state functions for the subsequent CI calculations. The CI calculations included 285 configurations with all the single and double excitations. The X ‘C potential curve for Na-Ne is presented in fig. 3; it is much more repulsive than the X ‘2 curve and also possesses a shallower potential minimum. However, this does not mean that both curves might cross at a certain sufficiently small internuclear separation. If so, this could be of importance for interpretation of some collisional effects observed in the electrondetachment process caused by collisions of Na- with neon gas [ 171. The present molecular calculations carried out in the internuclear separation range up to 1.5 au
I
5
10
I R la.u.1
Fig. 5. Interaction energy curves for the NaNe, Na +Ne and Na-Ne systems.
show no curve crossing between the investigated potential curves (see fig. 5 ).
5. Conclusions
A modified version of the pseudopotential calculations for the adiabatic potentials was applied for the NaNe, Na+Ne and Na-Ne complexes, The rcsults of the calculations indicate that some troubles with using the i-dependent pseudopotentials of rare gases in molecular calculations can be avoided. This might be done simply by taking a RG atom in its ground-state configuration (SCF determinant ) which is kept frozen. In this approach, heavier RG atoms might be treated as eight-electron systems. The effect of their core electrons could be simulated by the appropriate f-dependent pseudopotentials, which are available from the literature. However, the polarization terms have to be included in the interaction Hamiltonian to allow for correlation effects. Further 577
Volume 173,number 5,6 test
calculations
CHEMICALPHYSICS LETTERS
are necessary to confirm this
presumption.
Acknowledgement All calculations were carried out on the CRAY XMP/2 computer at the Max-Planck-Institut fur Quantenoptik in Garching. One of us (EC) wishes to express thanks to Professor K.L. Kompa for a MaxPlanck-Fellowship for the period 1 August-31 October 1989. The authors are also grateful to Dr. U. Wedig (Max-Planck-Institut fur Festkorperforschung, Stuttgart) for implementing the MELD program in Garching.
References [ I] E. Czuchaj, F. Rebentrost, H. Stall and H. Preuss, Chem. Phys. 136 (1989) 79.
[ 21 E. Czuchaj, F. Rebentrost, H. Stall and H. Preuss, Chem. Phys. 138 (1989) 303.
578
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[3] W. Mueller, J. Flesch and W. Meyer, J. Chem. Phys. 80 (1984) 3297. [4] L. McMurchie, S. Elbert, S. Langhoff and E.R. Davidson, MELD program, Washington University, Seattle, USA (1982). [S]T.H.DunningJr.,J.Chem.Phys.55 (1971)716. [61 H. Hotop and W.C. Lmeberger, J. Phys. Chem. Ref. Data 4 (1975) 539. [ 71 D.W. Norcross, Phys. Rev. Letters 32 ( 1974) 192. [8] C.E. Moore, NBS Circular No. 467, Vol. 1 (US GPO, Washington, 1949). [ 9 ] M.D. Havey, SE. Frolking, J.J. Wright and L.C. Balling, Phys.Rev.A24 (1981) 3105. [lo] SF. Boys and F. Bemardi, Mol. Phys. 19 (1970) 553. [ 1I ] W.P. Lapatovich, R. Ahmad-Bitar, P.E. Moslcowitz, I. Renhom, R.A. Gottscho and D.E. Pritchard, J. Chem.Phys. 73 (1980) 5419. [ 121J.H. Goble and J.S. Winn, J. Chem. Phys. 70 (1979) 2051. [ 131F. van den Bergand R. Motgenstem, Chem. Phys. 90 ( 1984) 125. [ 141G. Peach, Comments At. Mol. Phys. 11 (1982) 101. [ 15] M. Philippe, F. Masnou-Seeuws and P. Valiron, J. Phys. B 12 (1979) 2493. [ 16] R. Ahlrichs, H.J. BShm, S. Brode, K.T. Tang and J.P. Toennies, J. Chem. Phys. 88 (1988) 6290. [ 171D. Scott, MS. Huq, R.L. Champion and L.D. Doverspike, Phys. Rev. A 32 ( 1985) 144.
CHEMICAL PHYSICS LETTERS
19 October 1990
Semi-empirical calculation of the potential curves of NaNe, Na+Ne and Na-Ne * E. Czuchaj ‘, F. Rebentrost, H. Stall ’ and H. Preuss * Max-Planck-lnstitut fir Quantenoptik, Ludwig-Prandtl-Strasse
10.~8046
Garching near Munich, Federal Republic
ofGermany
Received 20 June 1990;in final form 2 August 1990
The ground-state potentials for Na+Ne, Na-Ne and NaNe as well as the A *Hand B ‘E potentials for NaNe have been calculated in the semi-empirical approach. The semi-empirical /dependent pseudopotential has been applied to simulate the effect of sodium core electrons, whereas the electrons of the neon atom have been kept frozen in the ground-state configuration. Polarization terms have been included in the interaction Hamiltonian to account for correlation effects. The SCF and valence CI calculations have been carried out in a Gaussian basis set. The calculated potentials are in good agreement with the experimental data and other calculations.
I. Introduction The results of our recent pseudopotential calculations on alkali-neon [ 1] and calcium-helium (neon) [ 2 ] systems exhibit an excessively repulsive character of all analyzed Z potential curves. It seems to us, one of the presumable reasons for this might be that the rare-gas atom pseudopotentials are still questionable. In this paper we present the results of a somewhat modified version of the pseudopotential approach. The interaction problem has been treated here in a non-symmetric manner, i.e. the sodium atom is considered as a one-electron system, because only its valence electron is treated explicitly. The effect of the core electrons is simulated by the corresponding semi-empirical I-dependent pseudopotential. Contrary to this, the neon electrons are kept frozen in the atomic ground-state configuration (the zeroth-order Hartree-Fock determinant). Due to this, the Coulomb and exchange interaction between the two atoms can be calculated directly. However, * Work supported partially by the Polish Ministry of National Education project CPBP 0 I .06. ’ Institute of Theoretical Physics and Astrophysics, University of Gdansk, Wita Stwosza 57,80-952 Gdadsk, Poland. ’ Institut fur Theoretische Chemie, Universitiit Stuttgart, Pfaffenwaldring 55, D-7000 Stuttgart 80, Federal Republic of Germany.
to allow for correlation effects in an efficient and simple way, we have incorporated the polarization terms in the interaction Hamiltonian. The use of the I-dependent pseudopotentials in molecufar calculations is described in detail in our previous papers [ 1,21, so it will not be repeated here. The present method is sketched in section 2. Details of the calculations are given in section 3 and the results obtained are discussed in section 4.
2. Method The calculation of the adiabatic potentials El(R) between the alkali atom (A) and rare-gas (RG) atom (B) in the Born-Oppenheimer approximation reduces to finding a solution to the Schriidinger equation for any internuclear distance R (in atomic units)
C&(R)
fl(I*,
rB, R) 9
(1)
where HA and Hz, are the Hamiltonians of the isolated atoms A and B, VABstands for the interaction between the two atoms, r,., and rz are the position vectors of the electron with respect to the alkali core and the RG atom, respectively, and R is the position vector of B with respect to A. In the present ap-
0009-2614/90/$ 03.50 Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
513
Volume 173, number 5,6
CHEMICAL PHYSICS LETTERS
preach, the Hamiltonian HA is the same as the corresponding one in ref. [ 11. On the other hand, we now take the Hamiltonian Ha as a ten-electron atomic Hamiltonian containing additionally the polarization terms. The latter are expressed as
v,(r,)= -
2B w4(rB,4 -
aq-6P, 7
B
W%B,6) ,
where lydand (Y,are the static dipole and quadrupole polarizabilities of the RG atom, respectively, and p, is the dynamical correction to (Y,. The cutoff function w is taken in the Gaussian form (3)
with the cutoff parameter 6. The interaction Hamiltonian VA, is defined as vABhj
rB,
R)
=V,,(r*,rB,R)+Vec(rA,rB,R)+Vcc(R),
(4)
where V,,describes the “pure” electron-electron and electron-nucleus interaction, I’,, involves all the socalled cross terms or three-body interaction terms and V,, represents the core-core interaction. All polarization terms include the cutoff functions defined as appropriate powers of eq. (3). More details on the one- and two-valence electron Hamiltonians used in the present approach are to be found in refs. [ 1,2]. Finally, the core-core interaction is expressed as 1/,(R) = K,(R)
-
Z:%
2~4
w4(R,6) -
Gff, 2R6
w6(R,4
where (Y,and S, stand for the dipole polarizability and the cutoff parameter of the alkali core, respectively, Z,, is the net charge of the alkali core seen by e- at an infinite distance (Z,= I ), and Z, is the RG atom nuclear charge (Z,=lO). In contrast to our previous calculations the short-range term I’,, encompassing the Coulomb interaction and exchange effects need not to be calculated here separately. It has been calculated directly within the SCF procedure. A detailed description of including an effective core-polarization potential in all-electron SCF and 574
valence CI calculations is given by Mueller and Meyer [ 31. In contrast with us, however, they did not use pseudopotentials and frozen atomic orbitals.
3. Calculations
(2)
w(r, 6) = 1-exp( -6r2)
19 October 1990
All the calculations have been performed with the MELD program [ 3 1. First, the SCF atomic orbitals for the ground-state configuration of the neon atom were obtained in the ( 5~3~) Gaussian basis set taken from Dunning [ 5 1. The Hartree-Fock energy for neon (polarization terms included in the interaction Hamiltonian) amounts to - 128.542587 au, which differs only slightly from the corresponding energy - 128.540448 au obtained by Dunning. Construction of the Gaussian basis set for Na, Na+ and Nawas based on the basis set reported by Mueller et al. [3] for the calcium atom. By dropping their eight largest exponents and adding two diffuse (0.0032 and 0.001) ones for s symmetry and, respectively, dropping six and adding two (0.0052 and 0.0021) exponents for p symmetry, we constructed the basis set for the negative ion of sodium. Finally, we took the (lOs9p3d/8s7p2d) basis set for Na- with the contraction coefficients -0.000474, 0.002696 and 0.007479 for s symmetry, 0.000045, 0.000201 and 0.000439 for p symmetry and 0.047965 and - 0.147777 for d symmetry. The calculated electron afinity for the (3s’) ‘S state of Na- amounts to 0.5453 eV, which agrees very well with the experimental value 0.546(5) eV [6]. For the (~P’)~P” state we have found the electron affinity to be 0.063 eV, which agrees with the theoretical value given by Norcross [ 7 1. Some of the exponents of the Na- basis set proved, however, insignificant in reproducing the experimental energies of a few lowest terms of the sodium atom. In consequence, in constructing the basis set for Na and Na+, we dropped the exponents 10.984, 0.2 and 0.00 1 for s symmetry and 1.763,0.0052 and 0.0021 for p symmetry. In addition, we added the exponent 0.008 for p symmetry. Finally, we used the (7s7p3d/%Sp2d) basis set with the contraction coefftcients 0.002189,0.014735 and -0.203912 for s symmetry, - 0.00399 1, -0.027910 and 0.082595 for p symmetry and 0.035585 and 0.078725 for d symmetry. The contraction involves the largest ex-
CHEMICAL PHYSICS LETTERS
Volume 173, number $6
ponents and was done to reduce the primitive Gaussian basis with slight degradation in atomic energies ( < 10m4au). The calculated ionization energies for a few lowest terms of the sodium atom along with the corresponding experimental values are compiled in table 1. The atomic basis set is, of course, too small to yield good agreement for higher terms. The size of the basis set for Na is motivated by the actual moIecular calculations. Namely, we wished rather to check whether the method works at all, and we were not interested in extensive molecular calculations. The only parameter needed to be determined in the present calculations is the cutoff one, 6, for neon (cf. eq. (2)). All the other parameters required in the calculations are taken from ref. [ 11. It has been found that the well-depth of the ground-state potential curve for NaNe depends very slightly on the cutoff parameter occurring in eq. (2). In contrast, the well depth of the A211 potential depends on 6 very strongly. Therefore, we decided to determine the value of 6 by matching the well depth of the A’II potential for NaNe to its experimental value determined by Havey et al. [ 91 from the temperature dependence of the intensity in the far-red wing of the Na resonance line perturbed by Ne gas. The value of 6 found in this way is 0.115 au. As can be seen in fig. 1, this value of 6 results in quite a realistic shape of the long-range interaction between a free electron and the neon atom. This is in drastic discrepancy with the long-range part of the pseudopotentials for rare gases used in our previous calculations [ 11, where the polarization terms are damped only at very short distances from the RG atom nucleus. The total pseudopotential (with its short-range term ) behaved, however, correctly. The same value of 6 has been used in the calculations on Na+Ne and Na-Ne. Table 1 Sodium-atom ionization energies (au) Term
(3s) ‘S (3P) *p (4s) ?s (4P) *p
Energy talc.
exp.
0.18881 0.11155 0.07155 0.05057
0.18886 0.11156 0.07158 0.05094
[81
19 October1990
Fig. 1. Long-range interaction between an electron and the neon atom defined by eq. (2) with the cutoff parameter 6= 0.1 I5 au.
As is well known, the so-called “basis-set superposition error” (BSSE) causes serious troubles in calculating interatomic interaction energies, especially if the basis sets are small. This problem has appeared in the present calculations as well. Our preliminary calculations on NaNe yielded the welldepth for the X ‘C potential to be much larger than its experimental value ( x 8 cm-‘). It has been found that the BSSE is responsible for this discrepancy. In the present approach (atomic orbitals of neon kept frozen), the molecular calculations were carried out in the sodium basis set augmented by the Is, 2s, 2p, 2p, and 2p, atomic orbitals of neon. In consequence, the SCF energy of the neon atom calculated in the molecular basis set weakly depends on the internuclear distance R. This insignificant effect for neon appears to be of great importance for the ground-state potential of NaNe. Representative values for the BSSE for NaNe are - 50, - 20 and - 5 cm-’ for internuclear distances of 5, 10, and 15 au, respectively, as obtained by subtracting the SCF energy of neon calculated in the atomic basis set from the corresponding SCF energy calculated in the molecular basis set in the interesting range of the internuclear separation [ lo]. Although the BSSE is quite negligible in the repulsive region of the X2C potential, it influences the attractive part of the potential very strongly and must be taken into account in the molecular calculations. 4. ResuIts and discnssion The reported molecular calculations have been 515
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performed for the NaNe, Na+Ne and Na-Ne complexes in the internuclear separation range between 1.5 and 40.0 au. Since the basis set used for sodium is insufficient to describe the Na terms higher than 3p with satisfactory accuracy, we restrict our discussion to the lowest X 2Z, A 211and B 2C potential curves. Numerical values of the calculated potentials are available upon request from one of the authors (EC). The calculated ground-state potential for NaNe is shown in fig. 2. The well depth of the X 2 potential (De=9 cm-‘) differs from that of our previous calculations [ 1] (D,=4.5 cm-‘) but agrees cm-‘) with the experimental value (D,=8.1{9) [ 111: The location ‘ofthe potential minimum on the R axis is, as in the case of our previous calculations (R,= 11.4 au) shifted from its experimental value (R,= 10.0 au) to the right by about 1 au. Our A ‘ll potential has been determined by matching the well depth of the calculated potential to its experimental counterpart. As seen in fig. 3, the potential minimum (De= 165 cm-‘, R,=5.3 au) is in good agreement with th’e experimental findings, although its location on the R axis seems to be slightly shifted from
19October 1990
-
I
1
I
300Na Ne
200 -
B2zz
A%
loo-
-i Si o‘;
-100-
I
5
.,
,
,
I,
IO
/
,
L
I,
I5 R (a.4
Fig. 3. The fmt excited potential curves for NaNe: A, Experimental value found by Havey et al. [ 91; 0, experimental value determined by Gable and Winn [ 121; 0, experimental value from refs. [ 11,131; 0, theoretical value obtained by Peach [ 141; I, calculated by Philippe et al, [ 151.
the experimental value to the right. In turn, the B 2 potential with its very shallow minimum (D,=3.0 cm-l) located at R,=15.3 au agrees with the experimentaldata (D,=3.0(5) cm-‘, R,=14.4(3) au) [ 111. In general, one has to state that the present potentials for NaNe are in better agreement with the experimental data than that of our previous pseudopotential calculations [ 11. It seems, however, that they are still too repulsive and the reason for this is as yet unknown. The SCF calculations carried out for the groundstate interaction of the Na+Ne system have yielded the potential curve depicted in fig. 4. The results of the SCF calculations supplemented by the dispersion term ( - C6/R6) yield the potential minimum, which lies near the theoretical minimum recently found by Ahlrichs et al. [ 161. Note that their calculations in-
Ok!++==I
5
I
I
I
I
I
10
I
I
I
l5 R(a.u.1
Fig. 2. Ground-state potential curves for NaNe and Na-Ne. A, Potential minimum for NaNe determined by Lapatovichet al.
1111. 576
19 octoba 1990
CHEMICAL PHYSICS LETTERS
Volume 173, number 5,6
0.4-
oc -
‘; ti 0.3.5 >
-2OO-
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; 5 0.2s
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a
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10
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15
Fig. 4. Ground-state potential curve for Na+Ne: - - - present SCF calculations; present SCF calculations plus -C6/R6 dispersion term taken from ref. [ 161; A, potential minimum found by Ahlrichs et al. [ 161.
elude the induction and dispersion terms of orders higher than Rm6.Finally, the SCF/CI pseudopotential calculations have been performed to obtain the ground-state potential curve for the Na-Ne system. In this approach, only two electrons of Na- were treated as the active ones. For each value of R, the SCF calculations were performed on the ground state of the system to obtain spatial orbitals. The latter were used to construct a basis set consisting of OTthonormal configuration state functions for the subsequent CI calculations. The CI calculations included 285 configurations with all the single and double excitations. The X ‘C potential curve for Na-Ne is presented in fig. 3; it is much more repulsive than the X ‘2 curve and also possesses a shallower potential minimum. However, this does not mean that both curves might cross at a certain sufficiently small internuclear separation. If so, this could be of importance for interpretation of some collisional effects observed in the electrondetachment process caused by collisions of Na- with neon gas [ 171. The present molecular calculations carried out in the internuclear separation range up to 1.5 au
I
5
10
I R la.u.1
Fig. 5. Interaction energy curves for the NaNe, Na +Ne and Na-Ne systems.
show no curve crossing between the investigated potential curves (see fig. 5 ).
5. Conclusions
A modified version of the pseudopotential calculations for the adiabatic potentials was applied for the NaNe, Na+Ne and Na-Ne complexes, The rcsults of the calculations indicate that some troubles with using the i-dependent pseudopotentials of rare gases in molecular calculations can be avoided. This might be done simply by taking a RG atom in its ground-state configuration (SCF determinant ) which is kept frozen. In this approach, heavier RG atoms might be treated as eight-electron systems. The effect of their core electrons could be simulated by the appropriate f-dependent pseudopotentials, which are available from the literature. However, the polarization terms have to be included in the interaction Hamiltonian to allow for correlation effects. Further 577
Volume 173,number 5,6 test
calculations
CHEMICALPHYSICS LETTERS
are necessary to confirm this
presumption.
Acknowledgement All calculations were carried out on the CRAY XMP/2 computer at the Max-Planck-Institut fur Quantenoptik in Garching. One of us (EC) wishes to express thanks to Professor K.L. Kompa for a MaxPlanck-Fellowship for the period 1 August-31 October 1989. The authors are also grateful to Dr. U. Wedig (Max-Planck-Institut fur Festkorperforschung, Stuttgart) for implementing the MELD program in Garching.
References [ I] E. Czuchaj, F. Rebentrost, H. Stall and H. Preuss, Chem. Phys. 136 (1989) 79.
[ 21 E. Czuchaj, F. Rebentrost, H. Stall and H. Preuss, Chem. Phys. 138 (1989) 303.
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[3] W. Mueller, J. Flesch and W. Meyer, J. Chem. Phys. 80 (1984) 3297. [4] L. McMurchie, S. Elbert, S. Langhoff and E.R. Davidson, MELD program, Washington University, Seattle, USA (1982). [S]T.H.DunningJr.,J.Chem.Phys.55 (1971)716. [61 H. Hotop and W.C. Lmeberger, J. Phys. Chem. Ref. Data 4 (1975) 539. [ 71 D.W. Norcross, Phys. Rev. Letters 32 ( 1974) 192. [8] C.E. Moore, NBS Circular No. 467, Vol. 1 (US GPO, Washington, 1949). [ 9 ] M.D. Havey, SE. Frolking, J.J. Wright and L.C. Balling, Phys.Rev.A24 (1981) 3105. [lo] SF. Boys and F. Bemardi, Mol. Phys. 19 (1970) 553. [ 1I ] W.P. Lapatovich, R. Ahmad-Bitar, P.E. Moslcowitz, I. Renhom, R.A. Gottscho and D.E. Pritchard, J. Chem.Phys. 73 (1980) 5419. [ 121J.H. Goble and J.S. Winn, J. Chem. Phys. 70 (1979) 2051. [ 131F. van den Bergand R. Motgenstem, Chem. Phys. 90 ( 1984) 125. [ 141G. Peach, Comments At. Mol. Phys. 11 (1982) 101. [ 15] M. Philippe, F. Masnou-Seeuws and P. Valiron, J. Phys. B 12 (1979) 2493. [ 16] R. Ahlrichs, H.J. BShm, S. Brode, K.T. Tang and J.P. Toennies, J. Chem. Phys. 88 (1988) 6290. [ 171D. Scott, MS. Huq, R.L. Champion and L.D. Doverspike, Phys. Rev. A 32 ( 1985) 144.