van der Waals interactions of II-state linear molecules with atoms. C6 for NO(X 2II) interactions
Formulas are derived for the van der Waals Cn coefficients for the interaction of a diatomic molecule in a II electronic state with an S-state atom. Two triatomic states arise from the degenerate IIstate. The average of the two energies has the usual Legendre polynomial (PL) angular dependence, but the difference in energies of the two states is shown to have associated Legendre polynomial (PML with M=2) angular dependence. Procedures for including spin orbit coupling are included, and the extension to interactions of Delta- and Phi-state molecules is discussed. Values of the spherical part of the C6 coefficients for the interaction of NO with He, Ne, Ar, Kr, Xe, H, Li, Na, K, Rb, Cs, H2, N2, O2, CO, CO2, and NO are obtained from frequency-dependent polarizability data using PadÃ© approximants. In addition, estimates of all the induction and angle-dependent parts of C6 are given for the NO-atom interactions.
van der Waals interactions of IT-state linear molecules with atoms. C6 for NO(X2n) interactions* Glen C. Nielson, Gregory A. Parker, and Russell T Pack Department of Chemistry. Brigham Young University, Provo. Utah 84602 and Group T-6. University of California Los Alamos. Scientific Laboratory. Los Alamos. New Mexico 87545 t (Received 23 October 1975) Formulas are derived for the van der Waals C, coefficients for the interaction of a diatomic molecule in a II electronic state with an S-state atom. Two triatomic states arise from the degenerate II state. The average of the two energies has the usual Legendre polynomial (PL ) angular dependence, but the difference in energies of the two states is shown to have associated Legendre polynomial (pr with M = 2) angular dependence. Procedures for including spin orbit coupling are included, and the extension to interactions of a. and <P-state molecules is discussed. Values of the spherical part of the C6 coefficients for the interaction of NO with He, Ne, Ar, Kr, Xe, H, Li, Na, K. Rb, Cs. H" N" 0" CO, CO2, and NO are obtained from frequency-dependent polarizability data using Pade approximants. In addition, estimates of all the induction and angle-dependent parts of C. are given for the NO-atom interactions. where (J = 0 and 1 give the even (+) and odd (-) states, respectively. However, as we have discussed elsewhere, 6 one has fects can be included afterward as discussed in Sec. IV. Thus, were it not for the interaction, the correct zeroth-order electronic states for the molecule, regardless of total spin S, would be denoted by IA,M,> = I 1, M.>, where A is the component of electronic orbital angular momentum along the molecular axis and M, is the component of electronic spin angular momentum S along the same axis. Since M. is being treated as a good quantum number at present, we will suppress it and write IA> for 1A, M.>. Now, when the atom is present, the only symmetry remaining, in general, is reflection in the plane of the three atoms, so that the degeneracy of 10 and 1- 1> is removed. If the x axis is kept perpendicular to the triatomic plane (Fig. 1), the proper zeroth-order molecular wavefunctions 1<7> are those which have definite parity (J under the operation a which reflects x into - x (but leaves spin space alone8); that is, I. INTRODUCTION When a diatomic molecule in II electronic state interacts with an S-state atom, the interaction breaks the degeneracy of the II state and two potential energy sur:" faces arise, one for which the electronic wavefunction is symmetric under reflection in the triatomic plane and one for which it is antisymmetric. In connection with our calculation of these two surfaces for the ArNO system1 using the electron gas model, we became interested in also determining the van der Waals potentials for such systems. This paper presents the results of our study. In the next section we derive the formula for the general second order van der Waals Cn coefficient for the nonrelativistic interaction of a II state diatomic with an S-state atom using the notation of our recent paper on those interactions for I:-state molecules. 2 Two surfaces are obtained. Others3 4 have combined these two surfaces by introducing dependence on an azimuthal angle; however, when one does that he must remember that the additional coordinate is an electronic coordinate and cannot be treated as a nuclear coordinate in collision studies. To avoid confusion, we stay strictly within the Born-Oppenheimer approximation and keep two surfaces, ( 1) (2) FIG. 1. Coordinate axes systems used for the interaction of a n-state molecule A (in this case NO) with an S-state atom B (such as Ar). The x axis of both systems is perpendicular to the plane of the three atoms. In Sec. n we also look in detail at the C6 coefficient and discuss ways to estimate the parts of it. In Sec. nl we apply these methods to the interactions of NO (x aII), obtaining the spherical part of C8 from frequency- dependent polarizability data using the Pade approximant methods of Langhoff and Karplus5 and esimating the angle-dependent parts of C6 from available data. Then, in Sec. IV we show how to include spinorbit coupling and also discuss the results. II. THEORY A. Derivation of general formulas We consider a neutral molecule A in a II electronic state and a neutral S-state atom B interacting at a distance large enough that electron exchange and overlap are negligible. In this section a completely nonrelativistic, spin-free Hamiltonian is used; spin-orbit ef- A I Z r 8 Ar The Journal of Chemical Physics, Vol. 64, No.5, 1 March 1976 Copyright @ 1976 American Institute of Physics 2055 2056 Nielson, Parker, and Pack: Interactions of linear molecules with atoms where (9) (10) j-2 ,c,JJ' =.!.I:L F(j, 1, m)F(j/,j' - j+ 1, - m)(- It'" 2 1 1 m E(Z} = - L r""Ck , k 6 where r is the distance from the center of mass of the nuclei of A to the nucleus of B, and in the notation of our recent wor on :E-state interactions,Copyright is given by (3) where S A is a natural parity dependent on phase conventions. (For our NO exa.mple, SA =0.) Similarly, if 10) is the ground (5) state wavefunction of atom B, it will automatically have some parity SB under fl, (It happens that S B =0 for all the atoms used as exampIes in thil:l paper.) With projection techniques one easily shows that the proper products to take as the two zeroth-order solutions of the triatomic problem are !u,O)= lu)lo)=2-1/2[il)+(-WA+SB+ol-l)]!0). (4) The long range potential is obtained by solving the electronic Schrodinger equation in the usual perturbative fashion. 7 The Born-Oppenheimer electronic Hamiltonian is written in the form, (5) XfanU -I-1,j -I-I, B). (11) where HA and HB are the electronic Hamiltonians of molecule A and atom B, and V is the electrostatic interaction. Because atom B has no permanent multipole moments, the first order interaction energy vanishes at large distance s, (6) Here F(j, 1, m) is a known coefficient, 2,7,8 the fan are the usual9 2i - I - 1 -pole oscillator strengths of atom B, and the QT(A) are the multipole moment operators of A (in atomic units), The second order energy can be written, in general, in terms of a Green's function as E(2) = - (0, a I(V -E(l))[HA +HB - Ea(A) - Ea(B) r1 X(V-E{1)la,O). (7) In evaluating this, one is free to use any spectral representation of the Green's function he wishes. For convenience we choose to use the complete set of functions Iv,n)= Iv)ln), where Iv) and In) are the eigenfunctions of A and B, respectively, which have definite !I. and ML(B) rather than definite reflection parity a. Then, because of the symmetry of B, Eq. (7) becomes E(2) = _LL' (0, u IV I II, n)(n, II IV lu, 0) (8) v n E:,,+ E:n ' where v =Ev(A) - Ea(A), etc., and the prime implies omission of the n= term. However, the II = I 1) induction terms are included. When the multipole expansion for V is substituted into Eq. (8), E(2) takes on the usual van der Waals form, The sum in Eq. (12) is over all nuclei and electrons belonging to molecule A with coordinates measured from the center of mass of the nuclei of A' and the z 1 axis taken to point along r as in Fig. 1. To simplify evaluation of the matrix elements in Eq. (11) we transform from the present coordinates (the primed set in Fig. 1) to a new set (unprimed in Fig. 1) in which the z axis points along the molecular axis R. In doing 50, the x axis is kept perpendicular to the plane of the three atoms so that reflection in the triatomic plane will be the same (x - - x) in both systems. To achieve this rotation through angle e about x using Euler angles, we rotate the axes by O! =371/2 around Z', then by f3 = e around the resulting y" axis, and then by 'Y =71/2 about the new z axis. The effect of this rotation on the multipole moment operators is I Q (r= z) =L D m(31T/2, e, 1T/Z)Q) (it = z), (13) JL _., where the Wigner D functions are the representations of of the rotation group. la, 11 This gives J-2 ( ) ( ' ) A =!."""" F( . I )F("" _. 1 _ )(_ 1)-"'''''' u 2 LJL.. J" m J ,J J +, m L..L.. D,I ,"n""i '-',i+-mIL".."L'.". 0' IQI I(v( v+IQ j'_) i+,1 0' J.O. n(J. - 1-1,J' - 1-I) , 1.1 m ,. ,.' ... n n v n (14) where we have surpressed the arguments of the D functions and also the A and B labels where v and n make them clear. Coupling the two Wigner D functions together via the Clebsch-Gordan theoremlO,12 gives If.;J Z"" . .1 "'" .1. 1 ') L ",,,,,(aIQ III)(vIQj:!+IIQ!) .. (. 1 ' ) AJI'=- LJO!(L,I,J,J)LJLJC(1,J -J+I,L;IJ,J.L,IJ.+IJ. D"+"',oLJLJ () JOnJ- -I,J-I-I, 2. L ,. .. ' ... n E" E:v + n (15) , a(L, I,j,i') =L (- 1)""'F(j,1, m)F(j',j' - j + 1, - m) C(l,j' - j +I;L, m, - m, 0), 1ft (16) (_I)J'_I+l+L [(2 i -l)I(2 j '-I)I]1/2.., . .1 "" (2j-21-Z)1 (21)I(Zj'-2j+21)1 C(J-l,J -1,L;OOO)W(;-l-l,I,J -I,L;J-l,J -J+l), (17) J. Chem. Phys" Vol, 64, No.5, 1 March 1976 Nielson, Parker, and Pack: Interactions of linear molecules with atoms 2057 and W is a Racah coefficient. 13 For those values of the parameters for which Q! is nonzero, it reduces to ( 1) <J-J'+L)/2-1(' ., L 2)I(j+j'+L-2) I .., _ - J+ J - - 2 r. (2L+1)(j-j'+L)lv' -j+L)! J1/2 Q!(L,1'J'J)-(2j_21_2)le+j';L-2)le-{+L)le' j+L)1 L(j'-j+21+1+L)!(j -j+21-L)! (18) Use of the definition [Eq. (4)1 for la), the m-type selection rule, and the fact that state Iv) has a well-defined angular momentum projection Av along the molecular axis allows the sums over IJ. and IJ.' in Eq. (15) to be done and gives /j,JJ'=1-L, .,L ,.,Ci(L,l,J.,J,),L J L',.f, on(j-1(-l,)j-I-1) [L'I QL,o C(1,J., -J.+ 1,L; 1 - A., A - 1, O)(lIQ1I-Av IV )(V IQAj'-vJ-1+I 11) 4 ' =1 L ." E" E.+ En x (V IQf. j!1 It> ]. (19) The D functions here reduce 10 ,14 to spherical harmonics or to associated Legendre polynomials, 15 D ,o (311"/2, e, 1T/2) =[41T/(2L + 1)J1/2y *( - e, -1T/Z), (20) [ (L-M)IJI/2 ..,M ) iN,/2 = (L +M)I , rdcose e . (21) Using this and the relation yt =(_l)My *, (22) to avoid dealing with p-i, we have A ....Jj.=1-L ,., L.-J(LCil"J',J.I,L,J L J,fon(j-(1-1),j-1-1){pLoC(Oes)rrC(,l"J-. J1 +L 1,;A-.A,.-1,O)(1I Q11-A.1 v ) 4 1=1 L " En + En x(V IQ ""--J ,I 1) +C(l,j' -j+1,L;-1- A., A.+ 1,0) (-1IQj l-a-lv)(vIQtr:l. , 1-1) J- (-1)8A8B+aG ::/:r/2 xpz (cos9)[C(1,j' -j+ 1,L;1- A., A.+ 1, 2) (1IQl-a-lv)(vIQtr_j ,1-1) +C(l,j' -j +1, L;-l- Av,Av -1, - 2) Thus, it is seen that in addition to the ordinary Legendre polynomial pl angle-dependent terms which are essentially the same as those obtained for the interactions of L;-state molecules,2 the natural expression for the Cn for TI-state molecules also involves the associated Legendre polynomials P . Since the P =P L form a complete set, one could express the pi as linear combinations of the PL , but, as we will show in a future paper1 on collisions in these systems, it is convenient to keep the pi in there. Also, the fact that P =0 when e=0 or 1T makes it clear that the difference in energies of the two states with different a vanishes, as it should, when the triatomic system is linear. From Eq. (23) one could generate explicit expressions for any Cn coefficient desired as we recently did in getting coefficients through Ca for -statemolecules. 2 However, there is presently not enough information available for most TI-state molecules to calculate anything past C6 with any reliability; hence, we only consider C6 in the next subsection. Before doing so, we note in passing that the procedure just presented can be extended very simply to treat van der Waals inter- (23) , actions of molecules in /j" ,etc. (A =2, 3, etc.) states. Everything goes through as before except that in Eq. (4) one has la) = 2-1/2[1 A) + (- 1)8ASs+a!_ A)]. (24) Equation (15) is obtained as at present, but Eqs. (19) and (23) are generalized, and the angle dependence obtained involves the PL 12AI in addition to the usual P . B. The C6 coefficient Let us now construct simpler explicit formulas for C6 We start from Eq. (10), C6 =/j,33' (25) and from Eq. (15), rather than Eq. (23), to make comparison with the usual formulas easier. Thus, C6 is given by 2 Ca= }' a(L, 1,3, 3)L) C(1, 1,L;/J,IJ.', /J+ /J') ,, J. Chern. Phys., Vol. 64, No.5, 1 March 1976 2058 Nielson, Parker, and Pack: Interactions of linear molecules with atoms We break this up into induction and dispersion contributions, The induction contributions come from the terms with Ill) = 11) and 1-1). Noting that -1"" /l, /l' "" 1, one sees that all matrix elements like (- 11Q 11) are zero due to the m-type selection rule, and all that survive are the /l= fl'=O terms, so that Ce(ind) becomes 2 Ce(ind) = L o.(L, 1,3, 3)C(I, 1, L;O, 0, 0)D0 0 o.(B) LoO and where (- 11Q IA.)(A" IQf 1- 1) = (11 Qj"l- A.)(- A.I Qj"'!I), (34) (- l1Q IA.)(A.IQn 1) =(I\Qj"l- A.)(- A.I Qi"'I-l>, (35) where a IA,,) = (- 1)"'Y 1- Av ). Now, the fl-type sums run symmetrically over positive and negative values, O!(L, 1,3,3) is only nonzero for even L, for which the Clebsch-Gordan coefficient is symmetric12 under change of sign of /l and fl', and in the present case _ o=Dt+""o, so that we can change the signs of fl and /.1' in the right hand sides of Eqs. (34) and (35). That proves the assertion for all states II for which A" =O. For any state with A. '* 0, there is always another state with the same energy and - A. in the sum, so that the total contributions are equal. Furthermore, we note that the m-type selection rule is only satisfied if Jl' = - Jl in the first term and Jl' = 2 - Jl in the third term of Eq. (33). Using these simplifications, we can write Ce(dis) =Ce(O, dis) + Ce(2, dis) p (cose) + (- l)aDeP (cose), (36) (28) (29) (27) O!(B) =L'!0. n-2, 1. Induction terms Ce=Ce(ind) + Ce(dis), and consider each separately. x [(lIQ Il)(lIQ 11)+(-IIQ I-1> x (- 11 Q 1- 1) ], where O!(B), given by where Here u only need reflect the molecular coordinates, and we have used the commutation relatione (39) (40) (38) (37) x2:'L: I (lIQlll1)(IIIQr 1- 1) !o.(l, 1). .( . + .) In the Ce(L, dis) we note that (1\ Q1'111) (III Qi" 11) = (- 1)" I (v I Qi 11) 1 2 , and define /l-dependent oscillator strengths2 by fo (I, 1)=2 .I(IIIQt101 2 where Ce(L, dis) = o.(L, 1,3,3)L:Copyright 1, 1, L; /l, - Jl, 0) " and for the atom fOn =fOn' To simplify the equation for the new coefficient, De, we note that the operators QI' and Qr" are both nonzero only if - 1 "" J.L .,,; 1and - 1 "" 2 - J.L "" 1 which is true only for Jl =1. This means also that the only states v that contribute to Eq. (38) are those with Av =OJ i. e., states. Using that, reflection symmetry, and Eqs. (39) and (40), one has With this it is clear that the Ce(L, dis), in which A= 1 is conserved, are exactly the well-known terms that appear in -state interactions. Evaluating the coefficients in Eq. (37) gives the usual formulas Ce(O, dis) =!L:" lOyff'M , (41) 2 .t. n .( n+ v) and Ce(2,diS)=-21 2:" !on (fg.-if .-tfo-.l), (42) .,. n .( n+ .) and De = (- 1)'A+'B+la (2, 1,3,3)L C(l, 1,2; /l, 2 - /l, 2)( 41 )"lf2 " (31) (30) (32b) uQ';' = Qj" IT (aIQ 11I)(IIIQ( !a) =i{[ (1IQ !1I)(IIIQ( 11)+ (-I!Qi Ill) x (1IIQi' 1-1)]+ (- l) A+3B....(I!Qi Ill) x (v]IQi' 1-0 + (- l!Qi Iv)(vlQf 10n (33) We assert that the first two terms here give equal contributions to the sum, and the last two do likewise. To prove this we represent Ill) by IA" > for clarity and use manipulations such as those in Eq. (30) to show that 2. Dispersion terms The dispersion contribution is given by the terms in Eq. (26) for which Ill) '" 1 1) and n '* O. The matrix elements involved are Use of Eq. (21) and evaluation of the coefficients in Eq. (28) gives Ce(ind) = Ce(O, ind)+ Ce(2, ind)pg (cosO), (32a) is the usual static dipole polarizability of atom B. The matrix elements left here are all equal to the ordinary dipole moment of A, due to /leA) = (lIQ 11) = (- 1)'A(u(-l) IQ 11) =(- l)'A( - 11 uQ? 11) = (- l)'A (- 11 Q u 11) = (- 11 Q 1- 1). which is exactly the formula obtained for - state molecules, so that nothing new is needed for this term for II states. J. Chern. Phys. Vol. 64, No.5. 1 March 1976 Nielson, Parker, and Pack: Interactions of linear molecules with atoms 2059 and or III. CALCULATIONS AND RESULTS a(w)= L S(-2k - 2) w2\ (51) k is expanded in the Cauchy series, molecule. However, since we use experimental data in the present calculations, the results represent an average over the ground state vibration of the molecule. Using the experimental dipole moment16 of NO, M(NO) =- 0.158 D= - O. 062 a. u. and the sum rules S(- 2) = cr(B) of Langhoff and Karplus, 5,19 we obtain the values of C6(L, ind) shown in Table n. . The frequency-dependent polarizability was obtained from refractive index data using the Lorenz- Lorentz equation,2D - 3 (if-I) Q' (w) = 41rn <?+2T (49) where T) is the refractive index at frequency w, and n is the number density of molecules. The BenedictWebb- Rubin equation of state 21 for NO gives 3/41Tn =60035 %at STP. Refractiveindexvalueswerefoundat 22 visible andultraviolet waveIengths ranging from 224. 7 to 670.9 run. 22.23 No infrared data were found; however, only the electronic (essentially the uv) part of a(w) goes into Eq. (46), and NO has a small dipole moment, so that it is clear from our work24 on CO that the ir spectrum of NO contributes negligibly to the available a(w). A rough value of the zero frequency polarizability of NO, a(O) =11. 70 O. 27 a , was obtained from ava ilable dielectric constant data2S and the Clausius- Masotti equation20 ; it is consistent with the value of a(O) =S( - 2) obtained below but not accurate enough to add any information. Because there is much less data available in this case than there was for C024 or CO2, 16 nothing could be gained by using the finite linewidth formulas employed there, and the Langhoff-Karplus5 procedure was followed directly. Briefly summarized, the frequency-dependent polarizability, a(w)= L:' , (50) Ev - W an the sum rules S( j) are determined by fitting the experimental data. Then, these sum rules are used to construct upper and lower bounding Pade approximants to a(w) which can be put in the form of Eq. (50) with a finite sum, and used in Eq. (46) to obtain bounds to Cs(O, dis). In fitting the limited available data with Eq. (51), five coefficients were kept and determined by an iterative procedure in which the first three coefficients were obtained from a linear least-squares method and S(- 8) and S(-10) were determined by a nonlinear method2B which assured satisfaction of the Stieltjes constraints. S The resulting sum rules, which fit the data with a standard deviation of O. 0068 a , are in Table I. The 8(0) in this table was obtained from the Reiche-Thomas-Kuhn9 theorem. To obtain reasonable uncertainty limits for the S(j), which reflect the effects of omitting the higher terms in Eq. (51), S(-10) (which is not needed in the [2,1] Pade approximants used) was varied within the range allowed by the Stieltjes COnstraints and the induced fluctuations in the SU) were (43) Cs(2, dis) =a6C6(0, dis), (47) where The van der Waals coefficients of the previous section are functions of the internuclear distance of the XL" (-l)s fo (1, 1)fon(1, 1)/(n(. (n+(.), (44) .,n ( all31l -Ol3/4)/( a"3/4 + 20l3/4) :Sas$IC= (o,,-ol)/(o,,+2al ), (48) and all and a l are the parallel and perpendicular polarizabilities of A, respectively. Evaluation of Eq. (45) to get Ds appears to be much more difficult. One might think of relating it to the similar parts of Cs(2, dis) but, in contrast to states, fo *fr;, in Eq. (42), and it should also be noted that the contributions of the excited!:' and - states in Eq. (45) have opposite signs. About all that can be said at the moment is that D6 should be small compared to Cs(O, dis). The only experiments3 4 that we know of, that do not involve the (J = and (J =1 states in equal numbers and give averages in which all information about Ds cancels out, require detailed analysis of collisions and will be discussed in a future paper. 1 It is now possible (but not easy) to obtain D6 from accurate ab initio calculations, and such calculations are to be encouraged. The equiValent calculations for interactions of P-state atoms have been carried out for some systems, and in the next section we use atomic results to get a rough estimate of Ds for NO interactions. Copyright Evaluation of C6 Now let us consider ways to calculate or estimate the values of the exact formulas [Eqs. (32), (41), (42), and (45)J. Accurate theoretical or experimental values of a(B) and J1.(A) are often available, so that the Cs(L, ind) are easily obtained for many systems. Cs(O, dis) can be obtained accurately from experimental frequency-dependent polarizability data using Pade approximants,5,la Eq. (41), and Ca(O, dis) = 3(21r)"1f aA(iy)aB(iy) dy, (46) . where aA(iy) is the average polarizability of molecule A at the imaginary frequency iy. If the frequency dependence of the polarizability anisotropy is known, then Cs(2, dis) can be obtained the same way. 17 However, if the polarizability anisotropy IC of A is known at only one frequency, then, as we have shown elsewhere, 2 reasonably good bounds on Cs(2, dis) can often still be obtained from ris=t(41)"1 /2(-1)SBO(2, 1, 3, 3)C(1, 1,2;1,1,2) J. Chem. Phys., Vol. 64, No.5, 1 March 1976 2060 Nielson, Parker, and Pack: Interactions of linear molecules with atoms IV. CONCLUSION which was used to generate the last column of Table II. The uncertainty in Eq. (55) is a guesstimate based on the following: the polarizability of NO is nearer that of C than 0; use of the rati029 for C would give 0 =0.035. But NO has more electrons than 0, so that changing the direction of one electron should have a proportionately smaller effec t and give a smaller 0 than thatfrom O. This uncertainty makes itclear that the present rough estimate of De is mostly an illustrative example calculation. A. Spin-orbit effects In the preceding sections we have assumed a completely nonrelativistic model with spin and orbital angular momentum completely uncoupled. For In molecules interacting with 1S atoms this is adequate. However, NO is a 2n-state molecule with spin-orbit splitting constant26 A=124.2 cm-I = 5.66 X 10-4 a. u. For the interaction of NO with the is atoms in Table II, the rdependent relativistic corrections are negligible, Sg so (54) (53) (55) + state to a 11. orbital which sticks out in the plane of the atom and is more strJngly polarized by it. This interpretation in terms of orbital directions also predicts correctly the relative values of all the L- and II-state polarizabilities calculated for P-state atoms by stevens and Billingsley. 29 Assuming that NO is like the 0 atom, that is, that the ratio of the two Ce coefficients in the direction perpendicular to the NO axis is equal to the ratio of 0 polarizabilities, 29 we have Cs(dis, 8 =11/2) _ ag(n) _ 0.66 Copyright(dis, 8=71"/2) - ag(L) -0.74 Letting De =OC6(0, dis) and using Eq. (36) we have 1-iae-30 _0.66 1- ae+31) -0.74' and with Eq. (52) we have 1) = 0. 018 0. 010, TABLE I. Oscillator strength sums for NO. Hartree atomic units. found. Because of the limited data available, these S(j ) are considerably less accurate than those obtained recently for other molecules. le,24 o 15.0000 -2 1l.518 0.013 -4 39.05 0.22 - 6 246 12 -8 (4.19 0.10)x103 -10 (1.0 0.I)x10s S (il The C6(0, dis) for the interaction of NO with a number of partners (using the sum rules of Refs. 5, 16, and 24) are in Table II. The values given are the means of the best Pade bounds, but the error bounds have been extended to include the effects of the uncertainties in the sum rules. Using the experimental value27 of the polarizability anisotropy of NO, K = 0.1617, and Eq. (48), we obtain a6 =0.141 0.021, (52) and use this and Eq. (47) to generate the Ce(2, dis) shown in Table II. To get a rough estimate of D6 we note that the contribution to S8 in Eq. (45) of closed shells is always zero, and all the atoms in Table II are either closed shell or have one s-type electron outside a closed shell, so that S8 =O. Also, since all the low-lying excited states26 of NO that occur in Eq. (45) are L+ states (s. =0), we expect De to be positive and from Eq. (36) Ce for the +(0'=0) state to be larger than that of the - state. This is consistent with the observation that the electronic structure of NO is basically that of closedshell N2eL;) plus one extra electron in a 71" orbital. The - state corresponds to a 7I"x orbital (see Fig. 1), and the TABLE II. Contributions to the van der Waals C6 coefficients for the interaction of NO (ZIT) with several partners. The angle-dependent terms are omitted for the molecular partners as the formulas in the paper are not appropriate for them. Hartree atomic units (e2 >. Partner C6(0. ind) =C6(2, ind) He Ne Ar Kr Xe H Li Na K Rb Cs Hz N7 C>.z CO2 CO NO 0.00535 0.0103 0.0429 0.0647 0.106 0.0174 0.633 0.647 1.11 1.22 1.38 9.8 1.3 21.0 3.6 69 10 98 15 170 31 20.2 1.4 177.8 4.6 217 16 321 25 364 30 314 16 28.2 2.4 71.1 9.3 55.2 4. 9 113 14 78 12 69 13 1.41 0.39 3.00 O. 95 9. 9 2. 9 14.0 4.3 24.3 8. 0 2.89 0.63 25.4 4.4 31.1 6. 9 46 10 52 12 44.8 8.9 9.8 1. 3 21.0 3.6 69 10 98 15 170 31 20.2 1.4 178.4 4. 6 218 16 3220125 365 30 315 16 1.41 0.39 3.0101 0.95 9.9012.9 14.1 4.3 24.4 8.0 2.91 0.63 26.1 4.4 31.7 6.9 47 10 53 12 46.2 8. 9 0.18 0.1l 0.38 O. 25 1.24 0.79 1.8 1.1 3.1 2.0 0.36 0.21 3.2 1. 9 3.9 2.3 5.8 3. 5 6.6 3. 9 5.7 3.3 J. Chern. Phys. Vol. 64, No.5, 1 March 1976 Nielson, Parker, and Pack: Interactions of linear molecules with atoms 2061 B. Discussion This has the eigenvalues E = (E + + E -)/2 {[(E+ - E-)/2]2 + (A/2)2P/2. (58) tific Laboratory through Subcontract No. XP5-72554. tpresent address of GAP and RTP. IG.Copyright Nielson, G. A. Parker, and R. T Pack, to be published. 2R T Pack, J. Chem. Phys. 64, 1257 (1976). 3K. Bergmann, H. Klar, and W. Schlecht, Chern. Phys. Lett. 12, 522 (1972), 4S. Green and R. N. Zare, Chem. Phys. 7, 62 (1975). 5p . W. Langhoff and M. Karplus, J. Chem. Phys. 53, 233 (1970); J. Opt. Soc. Am. 59, 863 (1969). 6R. T Pack and J. O. Hirschfelder, J. Chern. Phys. 49, 4009 (1968); 52, 521 and 4198 (1970). 7J. O. Hirschfelder and W. J. Meath, Adv. Chern. Phys. 12, 1 (1967). 8M E. Rose, J. Math. and Phys. (Cambridge, Mass.) 37, 215 (1958). 9J . O. Hirschfelder, W. Byers Brown, and S. T. Epstein, Adv. Quantum Chern. I, 255 (1964). 10T he D used here are those of Ref. 6. If one remembers that D (here) =Dt (Ref. 11) he can use the formulas of Ref. 11. 11M E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), p. 52. 12Reference 11, p. 58. 13Reference 11, p, 110. 14 Reference 11, pp. 54 and 60. Because of the difference in conventions (Ref. 10) this is diffeJ'ent from the usual reduction. !5Hanolbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965). 16R T Pack, J. Chern. Phys. 61, 2091 (1974). 17p . W. Langhoff, R. G. Gordon, and M. Karplus, J. Chern. Phys. 55, 2126 (1971). 18See F. P. Billingsley II, J. Chern. Phys. 62, 864 (1975) and references therein, !9Better values of SOme of these polarizabilities are in R. R. Teachout and R. T Pack, At. Data 3, 195 (1971), but the induction contribution is so small here anyway that they make no difference and the S( i) of Ref. 5 are all kept for the dispersion calculations because they form consistent sets. 20J O. Hirschfelder,Copyright F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954), pp. 861 and 883. 21 0 N. Seshadri, D. S. Viswanath, and N. R. Kuloor, J. Chern. Eng. Data 12, 70 (1967). 22Landolt-Bornstein, Zahlenwerte und Functionen (SpringerBerlin, 1962), 6th ed., VoL II, Chap. 8, pp. 871-901. 23International Critical Tables, edited by E. W. Washburn (McGraw-Hill, New York, 1930), Vol. VII, PP. 6-11. 24G. A. Parker and R. T Pack, J. Chern. Phys. 64, 2010(1976), 25C P. Smith and K. B. McAlpine, J. Chern. Phys. I, 60 (1933). 26D M. Himmelblau, ApPlied Nonlinear Programming (McGraw-Hill, New York, 1972), p. 458. 27N. J. Bridge and A. D. Buckingham, Proe. R. Soc. London Ser. A 295, 334 (1966). 28G Herzberg, Spectra of Diatomic Molecules (Van NostrandReinhold, 'New York, 1950), 2nd ed., p. 558. 29w. J. Stevens and F. P. Billingsley II, Phys. Rev. A 8, 2236 (1973). 30W. J. Meath, J. Chern. Phys. 45, 4519 (1966), 31G. Starkschall and R. G. Gordon, J. Chern. Phys. 54, 663 (1971). (57) (56) *Work performed in part under the auspices of the USERDA and supported in part at Brigham Young University by the USERDA and the University of California Los Alamos Scien- In this paper we have rigorously shown that the van der Waals potential between a II-state molecule and an S-state atom does have the associated Legendre polynomial dependence which others have assumed3 or argued4 that it should have. The accuracy of the present C6(O, dis) for NO interactions is the best obtainable using the present data; other methods17 would give the same results from the same data. However, it might be possible to obtain improved accuracy by using the method of Starkschall and Gordon31 which allows one to also use other types of data. A better determination of the new D6 coefficients than the present rough estimate is likely to be difficult experimentally and best accomplished by accurate ab initio calculations. that all that need be added is the spin-orbit splitting of the NO. This can be done by using the phenomenological Hamiltonian, For small r with 6 not near 0 or 11 the E+ - E- term dominates and the present basis is appropriate. For large r or enear 0 or 11 and the LS coupling dominates and the appropriate basis is the set IlL, Ms' 0), which describes the 2II3/2 and 2II I/2 states. In the present examples, the spin-orbit coupling dominates in most of the van der Waals region, and the appropriate potentials are easily generated using Eq. (58). The above approach can also be used for the interaction of NO(2II) with the 2S- s tate atoms of Table II unless very high accuracy is desired in which case one would need to include the small ( O. 05 em-I) magnetic dipoledipole interaction. 30 instead of Eq. (5). The two terms in this H are not simultaneously diagonable, so that if we use the eigenfunctions of He obtained in the previous sections (denoted by I a, 0) '= I a, Ms' 0) = I , M., 0) with eigenvalues E*) as a basis, a two by two secular equation results (only 2 x 2 as Ms stays a good quantum number here) which has the simple form J. Chern. Phys., Vol. 64, No.5, 1 March 1976