J. Phys. Chem. 1982, 86,1099-1102
1099
Long-Range Interactions of Na(3s 2S) with Na(3s *S) or Na(3p *P) Danlel D. Konowalow' and Marcy E. Rosenkrantz Department of Chemktty, State UnIvers#y of New York at Binghamton, Binghamton, New York 13901 (Recelvd: June 22, 1981; In Flnal Form: October 23, 1981)
The potential energy curves for the eight lowesblying electronicstates of Na, which correspond to the interaction of Na(3s with Na(3s or Na(3p are calculated ab initio for internuclear separations in the range 15 IR I 30 bohr radii. The binding energies are analyzed to obtain estimates of the exchange energies, first-order electrostatic energies, and second-order dispersion energies.
Introduction Recently we have reported ab initio all-electron quantum mechanical investigations of eight low-lying electronic states of Liz and Na2 and of six low-lying electronic states of Liz+and Na2++ .'' These computations were carried out by using the multiconfiguration self-consistent field (MCSCF) scheme as implemented by Das and Wahl.lo The aim of this approach is to account for only that correlation energy which changes as a function of internuclear separation. The separated atoms are described a t the single configuration self-consistent field SC-SCF level in the Hartree-Fock-Roothaan scheme. Our earlier computations on N a z encompassed internuclear separations of R < 15 bohr radii. We have found that, in order to attain our present goal of reasonably accurate long-range interaction energies, it was necessary to double the fineness of the numerical integration grid in the 5 and r] coordinates which we had used before. See ref 10 for details on the numerical integration procedures. Here, we use the same set of Slater-type basis functions which we had used for the investigations at short range.6 The basis set listed in Table I is designed to describe the Na(3s) and Na(3p) atoms at essentially the Hartree-Fock level and to describe molecular polarization effects at medium range adequately. Thus, we expect it to be adequate to obtain reasonably reliable potentials at short and moderate separations. However, the basis set was not optimized to provide the best possible description of all aspects of the 3s-3s and 3s-3p interactions at long range. We expect the dipole polarizability of Na(3s to be accommodated quite nicely by the present basis set which contains three 3p functions in the I: symmetry block. The ll projection of the dipole polarizability of Na(3p 2P)will be described moderately well by the three d-type functions in the II block. Since we retained only a single unoptimized d function in the I:block for reasons of economy, the 2 projection of the dipole polarizability of Na(3p 2P)and (1)M. L. Olson and D. D. Konowalow, Chern. Phys. Lett., 44, 281 (1976). (2)M.L.Olson and D. D. Konowalow, Chern. Phys., 21,393 (1977). (3)M.L.Olson and D. D. Konowalow, Chern. Phys., 22, 29 (1977). (4)D. D. Konowalow and M. L. Olson, J. Chern. Phys.,67,590(1977). (5)D. D.Konowalow and M. L. Olson, J. Chern. Phys.,71,450(1979). (6)D.D. Konowalow, M. E. Rosenkrantz, and M. L. Olson, J. Chern. %ye., 72,2612(1980). (7)D. D. Konowalow and M. E. Rosenkrantz, Chern. Phys.Lett., 61, 489 (1979). (8)D. D. Konowalow, W. J. Stevens, and M. E. Rosenkrantz, Chern. Phys. Lett., 66, 24 (1979). (9)D. D. Konowalow and M. E. Rosenkrantz, ACS Sympcaium on High Temperature Chemistry, 1981. (10)G. Daa and A. C. Wahl, BISON-MC: A FORTRAN Computing System for MCSCF Calculations on Atoms, Diatoms and Polyatoms, Argonne National Laboratory, Report ANL-7955,1972. 0022-3654/82/2086-1099$01.25/0
TABLE I: STO Basis Set for Na," n I m b n
I
m
b
f
0 0 15.30 2 1 1 7.72 0 0 10.10 2 1 1 4.18 0 0 6.26 2 1 1 2.27 3 1 1 1.20 0 0 2.98 3 1 1 0.80 0 0 1.24 3 1 1 0.60 0 0 0.80 0 0 0.75 3 1 1 0.40 1 0 7.72 3 2 1 1.30 1 0 4.18 3 2 1 0.75 3 2 1 0.40 1 0 2.27 3 2 2 1.40 1 0 1.20 1 0 0.80 3 2 2 1.00 1 0 0.50 3 2 2 0.70 3 2 2 0.35 2 0 0.83 a This set of functions is placed on each atom. The m value indicates the molecular symmetry in which these functions are used.
1 1 3 2 3 3 4 2 2 2 3 3 3 3
TABLE 11: Configuration Lists for MC-SCF Wave Functions" state
configurations
X 'zg+ [ ](417,' t 40,' t 5ug2 + 50,' + 60 ' t 60,' t 2ng + 2n,' t 3ng' + 3n,' t l f i g z g + 16,Z) x 3zu+ [ 1(4ug4a, + 5ug5uu t 2ng2n, t 16g16,) A 'Xu+ [ ](40g5u, t 4u,5ug + 2ng2n, + 16g16,) b 3zg+ [ ](4ug50g t 4 ~ ~ 5 + 0 ,2nu3n, + 2ng3ng) a 3nu [ l(4ug2n, t 4u,2ng t 5ug3nU) B 'nu [ ](4ug2n, t 40,2ng + 5og3n,) C Ing [ ](4ug2ng + 4uu2n, + 5ug3ng) c 'rig [ ](4ug2ng t 4uu2n, + 5ug3ng) " The brackets [ ] represent the core electron configuration [ l u g '10, '20 '20, 2 3 ~ g 2 3'lnU4ing4]. ~U
the quadrupole polarizability of Na(3s are both relatively poorly described. The list of configurations we used in our variational computations is given in Table 11. Because of certain limitations of the MC-SCF code we used, we do not consider the pair of states 2 lZg+ and 2 32u+ which correspond asymptotically to the interaction of Na(3s 2S)+ Na(3p P). For the present computations at long range, the wave function is comprised almost entirely of the first two configurations which are listed for each state. These are the two configurations which are needed to assure dissociation to the proper pair of Hartree-Fock atoms. The reason is easy to see. Consider the 311, state, for example. The simple molecular orbital (MO) description of this state is (lu~ls)z(lauls)2(2ug2s)2(2uu~s)z(3u~2p)z(3uu2p)2( 1 ~ ~ (1~~2p)~(4~,3~)'(2?r,3p)', 2 ~ ) where triplet coupling of the valence electron spins is understood. Clearly, this description lists the product of the diagonal elements of the Slater determinant used to describe the MO. This MO 0 1982 American Chemical Society
1100
The Journal of Physical Chemistty, Vol. 86, No. 7, 1982
Konowalow and Rosenkrantz
TABLE 111: Binding Energy Curves" for Various Electronic States of Na, R, bohr radii lzg+ 3zu+ Izu+ 3zg+ 3%1 In, 3% 'nu 15.0 -0.000 234 -0.000 185 -0.005 561 -0.004 425 -0.002 396 -0.002 055 tO.001898 + 0 . 0 0 1 7 5 4 18.0 -0.000 068 - 0 . 0 0 0 0 6 4 -0.002 605 -0.002 458 -0.001 233 -0.001 197 + 0 . 0 0 1 1 2 3 tO.001105 21.0 -0.000 024 -0.000 023 -0.001 513 -0.001 491 -0.000 750 -0.000 745 +O.OOO 717 tO.000 715 24.0 -0.000 009 - 0.000 009 - 0.000 987 - 0.000 989 -0.000 495 -0.000 494 t 0.000 485 t 0.000 485 27.0 -0.000 003 -0.000 003 -0.000 689 -0.000 690 -0.000 345 -0.000 345 tO.000 343 tO.000 343 30.0 -0.000 000 - 0.000 000 - 0.000 501 - 0.000 501 -0.000 250 -0.000 250 t 0.000 252 t 0.000 252 " Energy in hartree atomic units relative to the asymptotic (R= - ) energy. For the lzg+and 'xu+ states R ( - ) = -323.716955 hartree; for 'xu+and 32g+ R ( - ) = - 323.644403 hartree; for the n states R(-)= -323.644406 hartree, where 1 hartree = e2/aois the Hartree unit of energy. TABLE IV: Components of the Long-Range Binding Energies
R values used, a, states '2 +
3 2
12:+:
3,;+
l%l,
+
3ng
'ng,
" C, = 164000 e2/ao.
A , e'la, a, ao-' C,, e2 ao2 C,, e 2aos footnote 0.345 0 1680 0.649 a 0.578 2.00 13.3 5800 b 2100 b 6.86 4.04 0.724 0.649 2900 b 6.65 2.47 CF = 2Ca = 2(6.31) e' ao2from King and Van Vleck.12
for A , a
for C,
15-21 15-24 15-21 15-2 1
15-21 24-30 18-27 21-30
may be abbreviated [](4ug)(2qJ,where the square brackets [ ] represent the core orbitals and ( )( ) represent the two valence orbitals. At large internuclear separations this single MO describes a combination of neutral atoms, a cation, and an anion. In this particularly simple case, only one other configuration needs to be added in order to introduce the left-right correlation which will assure dissociation to neutral atoms; it is [ ](4a,)(21rg). (Is it not curious that a configuration which is comprised of two antibonding valence orbitals is needed to lower the energy of the system sufficiently to assure proper dissociation?) The smallest collection of configurations which assures proper dissociation is called the base configurations. The dissociation is proper only in a nonrelativistic sense, however, since we ignore the spin-orbit coupling which is responsible for a fine structure Splitting" of 17.2 cm-' in the Na(3p 2P) state.
Results The results of the present binding energy computations are listed in Table 111. They are shown in Figure 1 together with other potential curves which were reported previously'+ in order to put the present work into proper context. It is quite obvious from Table I11 and Figure 1 that pairs of potential curves have very nearly identical long-range behavior. If one of the pair has a lAi term symbol then the other of the pair has a 3Aj symbol. Here the subscript i denotes the inversion symmetry g (gerade) or u (ungerade); if i = g then j = u and vice versa. It is evident that the potentials corresponding to the ns + np asymptote have a substantially longer range than those corresponding to the ns + ns asymptote, and the excited Z states have a long-range interaction about twice the magnitude of that of the II states. The forms of the long-range interactions are relatively well
3
6
9
12 15 18 R (BOHRS)
21
24
Flgure 1. Low-lying electronic states of Na, and Na,'.
Let us assume that the binding energies (BE) of the lowest IZg+ and 32u+ are of the form
BE: = -C6R4 - C,R-s
f A exp(-aR)
(1)
at large separations. The inverse power terms are the leading terms of the dispersion interaction energy expansion and the exponential term is expected to be a crude but reasonable representation of the exchange interaction. 17,18 (Clearly, the plus sign in eq 1corresponds to the triplet state.) Knox and Rudge,Is for example, have used the form AV(R) = ARB exp[(ao/R) - a'R]
(11)C.E. Moore, Natl. Bur. Stand. Circ., No. 467 (1949). (12)G.W. King and J. Van Vleck, Phys. Reu., 55, 1165 (1939). (13)B. Linder and J. 0. Hirschfelder, J. Chem. Phys., 28,197(1958). (14)R. S.Mulliken, Phys. Reo., 120,1674 (1960). (15)J. 0.Hirschfelder, C. F. Curtias, and R. B. Bird, 'Molecular Theory of Gases and Liquids", Wiley, New York, 1954. (16)A. Dalgarno in "Advances in Chemical Physics", J. 0. Hirschfelder, Ed., Vol. XII, Wiley, New York, 1967,pp 143-64. (17)E. A. Mason and L. Monchick in "Advances in Chemical Physics", J. 0. Hirschfelder, Ed., Vol. XII, Wiley, New York, 1967,pp 329-87.
27
(2)
to represent the splitting at large separations of the lowest-lying singlet and triplet potentials of alkali diatomic molecules. However, since we have insufficient data from which to parametrize such a representation, we use the simpler exponential of eq 1. (18)H. 0. Knox and M. R. H. Rudge, Mol. Phys., 17, 377 (1969).
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982 1101
Lowest-Lying Electronic States of Na,
A least-squarea fit of the sums and differences of the binding energies at long-range gives the parameter values listed in Table IV. Clearly, we could use only the data for 15,18, and 21 bohr radii in the analysis because of the loss of significant figures when we analyzed the binding energies at larger separations. Since our basis set was not designed to describe the quadrupole polarizability of the 3s atom accurately, and since higher order inverse power terms are missing from eq 1,the C8 value we deduce is not expected to be particularly accurate. By contrast, our value of c6 is about 6% larger in magnitude than the value estimated by Dalgarno16 by entirely different means. The 'Z +-3Zu+ energy splitting which can be deduced from Tabfe I11 is generally somewhat smaller than, but in reasonable accord with, that computed by Knox and Rudge.18 The potentials corresponding to the Na(3s) + Na(3p) asymptote provide a wealth of information on long-range interactions. King and Van Vleck12 showed that the fmt-order resonance interaction between Na(3s) + Na(3p), for example, would give an R-3 energy dependence. Later w o r k e r ~ ~extended ~J* their work and corrected some errors in the original. In addition, the interaction potentials we treat are expected to have an attractive, R4 dependent dispersion interaction term (aswell as terms proportional to higher powers of R-l) and an exchange term. We approximate the long-range energies as follows: In,, 311gBEE = +CfR-3 - CfR4 f A exp(-aR) (3) lug, 311uBE; = -C3"R-3 - CfR4 f A exp(-aR)
(4)
lZU+,32,+B& = -CfRb3 - C#R4 f A exp(-aR) (5) where the triplet states have the repulsive exchange energy. Recall that the reason we do not treat here the 21Zg+, 2 lZU+pair is due to program limitations. We have confirmed in other computations that the long-range form of this pair represented approximately by 2 lXg+,2 3Zu+ = +CiR-3 - C;R+ f A exp(-aR). Let us deduce the constants which characterize the potentials for each pair of states in turn. By averaging the binding energies for the pair of states ln,, 311gas given in eq 3 we obtain the inverse power portion of the potential. A plot of R3 times the average binding energy vs. R3for the values 21 < R < 27 bohr radii gives a straight line with an R3 0 intercept of C" = 6.86 e2a,,2. King and Van Vleck12obtained the value C i = 6.31 e2 ao2 from the experimental oscillator strength for the 2S-2Ptransition. We obtain the value C t = 2100 e2ao5 in the same analysis. King and Van Vleck estimated roughly the value c6 = 3520 e2 a,,5 by fitting the Rydberg-KleinRees (RKR) potential for the high-lying vibrational levels of the 'nu state. We know now that the data they used to deduce a C t value corresponded to internuclear separations somewhat smaller than about 10 bohr radii where -C@ is expected to be an inadequate description of the interaction potential. Since the effect of the attractive exchange energy as well as terms like -CJC8 are lumped into their c6 value, it should be too large in magnitude. So it appears that our values are in reasonable accord with experiment. The exchange energy for the In,, 311gpair is estimated by a least-squares fit of the differences in binding energies for 15 5 R I21 bohr radii. (Our differencing technique results in substantial loss of significant figures for separations R > 21 bohr radii as is evident from Table 111.) The 'II ,311upair of states was treated similarly. A plot of R3[BEflII ) BE(311,)]/2 as a function of R-3 yields Cf = 6.65 e2 ao2'as the intercept and Cy = 2900 e2 ao5 as the
-
+
slope. Such a plot for the lZ,, 3Z9+pair yields C,"= 13.3 e2 a,,2 and Ct = 5800 e2 ao6. These values are twice the magnitude of those determined from the lIIg, QU pair as demanded by the0q7.l~'~(The CF value deduced from the 'nu,314pair appears to be too high by about 3%. We will look into possible sources of this discrepancy shortly.) Our preferred value Cy = 6.65 e2ao2 is about 5% higher than the value Cy = 6.31 e2 ao2 determined by King and Van Vleck12 from the relation c3
= fij/.ijgj
= p2
(6)
where fij is the experimentally determined atomic oscillator strength [which is judgedm to be in error by no more than 3%], vijis the frequency of the atomic resonance transition, gj is the degeneracy of the lower state, and p is the atomic transition dipole moment. We have found21 that our transition dipole moment functions calculated at large separations for molecular transitions corresponding to the atomic resonance transition agree within 2.5% with the experimental values: This is consonant with the 5% discrepancy in the C3 values as is seen from eq 6. Let us reconsider the discrepancy between the C3values we deduced from the two pairs of II states. One possible source of the discrepancy is the way we fix the asymptotic energies. Note in Table I11 that we have listed two different asymptotic energy values for states corresponding to a 2Sand 2p pair of atoms. We determine the asymptotic energies in molecular calculations in which the value of all two-center integrals were set identically equal to zero. The asymptotic energies reflect the constraints imposed by the cylindrical molecular symmetry. Consequently, the asymptotic energies of the excited Z states is different by some 0.66 cm-' from that for the II states and is different from the value we would have obtained by using this same basis set in calculations of the separated (spherically symmetric) atoms. The excact fixing of the asymptotic energy affects neither our determination of the transition dipole moment21nor of the exchange energy but it affects quite sensitively our determination of the inverse power terms. For example, a difference in asymptotic energy of 10" e2/% makes only a trivial (2.2 cm-l) difference in our estimate of the dissociation energy, but it can have a profound effect on the value of the coefficients of the inverse power terms we use to describe the long-range potential. It is clear that if we were to raise the 3s + 3p II asymptote by, say, hartree and lower the Z asymptote by about half that amount, we would improve the consistency of our Cg and Cf values to better than 3% (by lowering Cg more than we lower C f ) and also improve substantially the agreement between our C3 values and those determined either experimentally or computationally from transition dipole moments. All of this is merely to illustrate that errors in computing integrals or in achieving a converged MC-SCF computation that would ordinarily be considered rather small can have a rather large effect on the long-range analysis of the kind carried out here. It appears that our long-range analysis may be used to measure the success or failure of the basic premise of the OVC version of the MC-SCF approach.22 The premise is that an accurate potential curve may be obtained by describing the separated atoms at the Hartree-Fock level and describing electron correlation in an MC calculation (19) D.D.Konowalow and J. L. Fish, unpublished calculations. (20) W. L.Wiese, M. W. Smith, and B. M. Miles, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (US'.), No. 22 (1969). (21) D. D. Konowalow, M. E. Rosenkrantz,and D. S.Hochhauser, to be published. (22) A. C. Wahl and G. Das, Adu. Quantum Chen., 5 , 261 (1970).
1102
J. Phys. Chem. lS82, 86, 1102-1106
to A. C. Wahl for providing us with copies of BISON and used in the binding energy computations, and to M. L. Olson for use of some of his datahandling routines. As a graduate student, D.D.K. had the pleasure of being infected with Joe Hirschfelder's enthusiasm for understanding long-range interactions betweeen atoms and molecules. It is inspiring to see that Joe is still as enthusiastic as ever and is contemplating new solutions for unsolved problems in molecular theory.
in which only excitations from the valence shell of electrons is considered. That we agree moderately well with C3 values deduced from both calculated and experimental transition moments (when such agreement depends so sensitively on so many aspects of the calculation) suggests that the OVC premise is basically sound.
BISON-MC programs
Acknowledgment. Thanks are due to G . Radlauer for help with the computations of the long-range parameters,
A Klnetlc Theory for Polymer Melts. 3. Elongational Flows R. Byron Bhd,' H. H. Saab,+ and C. F. Curtisst Department of Chemicel E n g M n g and Rheokgy Research Center, and Department of Chembby and Theoretical Chembtry InstiMe, Unlversky of Wlsconsln-Mdson, Madison, Wlsconsln 53706 (Received: July 2, 1981; In Final Form: September 15, 1981)
The Curtiss-Bird constitutive equation, obtained from a reptation model for polymer melts, is applied to elongational flows. The steady-state elongational viscosity and the elongational growth function are calculated; it is found that these properties are very sensitive to the value of the link tension coefficient. The theoretical calculations are compared with the limited experimental data available.
Introduction In two recent publications Curtiss and Bird's presented a kinetic theory for undiluted polymers, based on the earlier Curtiss-Bird-Hassager phase-space In this polymer melt kinetic theory the macromolecules are modeled as freely jointed bead-rod chains (the Kramers "pearl-necklace model"), and the "reptation" concept used by Doi and Edwards5is introduced to account for the constraints imposed on a macromolecule by its neighbors. The Curtiss-Bird theory contains several parameters describing the macromolecule: N , the number of beads in the chain; a, the length of a rod connecting two successive beads; {, a bead friction coefficient; E , the link tension coefficient; and p, the chain constraint exponent. Some of these parameters occur together in the time constant, which arises in the derivation of the equation for the orientational distribution function: X = N3+s{a2/2kT. The Curtiss-Bird theory gives an expression for the stress tensor in terms of several kinematic tensors (i.e., a "rheological equation of state" or "constitutive equation"). For e = 0 and = 0 the constitutive equation is very similar to that of Doi and E d ~ a r d s .These ~ authors computed a number of the rheological functions for shear and elongational flows, and it is generally accepted that their results are qualitatively in agreement with experimental observations; they did not, however, present any data comparisons. In this paper we develop expressions for the elongational properties from the Curtiss-Bird theory and compare the calculated results with available experimental data. In the next paper in the series we do the same for shear properties. The Constitutive Equation The constitutive equation from the Curtiss-Bird theory is an expression for the stress tensor ?r = p6 + T in terms t Department of Chemical Engineering and Rheology Research Center. *Department of Chemistry and Theoretical Chemistry Institute.
0022-365418212086-1102$01.25/0
of two kinematic tensors: i. = Vv + ( V V ) and ~ ~[O](t,t?. Here T is the total stress tensor, 6 is the unit tensor, 7 is that part of the stress tensor that vanishes at equilibrium, and ( V V ) ~is the transpose of Vv. The tensor y[OI(t,t') is a finite strain tensor defined else where;'^^ it contains information about the complete history of the deformation of a fluid element. The Curtiss-Bird constitutive equation is 7
= NnkT[y36 -
in which p and
u
st -m
~ (- tt?A(t,t? dt'-
are given by
and the tensors A and B are
A = uu[l
+ y[ol:uu]-3/2
B = '/,Xi.(t):uuuu[l+ ~ [ O ] : U U ] - ~ / ~
(4)
(5)
Here the overbars indicate averages over a unit sphere (...) = (1/4a)So2"Jo"( ...) sin 1!9 d0 d@,and u is a unit vector. The above constitutive equation differs from that of Doi and (1)C. F. Curtiss and R. B. Bird. J. Chern. Phvs.. 74.2016-25 (1981): Errata: In (5.6)change [uj]to [uj];in (6.2)change gvt; to x v t u ; in the first line of (A51 change - to +. (2)C.F. Cur& an8 R. B. Bird, J. Chern. Phys., 74,2026-33 (1981); Erratum: In (4.2)uuuu should be uuuu. (3)C.F.Curtiss, R. B. Bird, and 0. Hassager, Adu. Chern. Phys., 35, 31-117 (1976). (4)R. B. Bird, 0. Hassager, R. C. Armstrong, and C. F. Curtiss, "Dynamia of Polymeric Liquids",Vol. 2, 'Kinetic Theory",Wiley, New York, 1977,Chapter 14. (5)M. Doi and S. F. Edwards, J. Chern. SOC.,Faraday Trans.2, 74, 1789-1801, 1802-17,1818-32 (1978);75,38-54 (1979).
0 1982 American Chemical Society
