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Chemical
Physics
ELSEVIER Chemical Physics 219 (1997) 161-
Inclusion of ion-pair states in the diatomics-in-molecules
description of potential energy surfaces: van der Waals complexes
of He-C12 and Ar-C
B.L. Grigorenko a, A.V. Nemukhin a, V.A. Apkarian b,*
a Department of Chemistry, Moscow State University, Moscow 119899. Russian Federation b Department of Chemistry, University of California, Irvine, CA 926192-2025, USA Received 14 January 1997
Abstract
It is shown that the inclusion of excited ionic configurations in the diatomics-in-molecules (DIM) Hamiltonian serves as a natural means to account for main features of non-additivity in three-body potential energy surfaces of He-CI 2 and Ar-CI 2 van der Waals complexes. For ground state C12(1Eg), while consideration of only neutral configurations leads to T-shaped isomers, inclusion of the excited CI+C1 - configuration stabilizes the linear isomer and destabilizes the T-shaped isomer. Within the same formalism, the excited C12(3I]) only sustains minima in the T-shaped isomer. Potential energy surfaces created with a minimal DIM basis are constructed and shown to compare favorably with the most accurate ab initio surfaces and experiments. Analytical forms are given for the three-body surfaces, meant for fitting purposes and as a convenience in simulations of dynamics. © 1997 Elsevier Science B.V.
1. Introduction
The aim of this work is to study the effect of including ionic configurations on the diatomics-in-molecules (DIM) construct of potential energy surfaces for van der Waals complexes. We will take as our specific examples the HeC12 and ArC12 complexes, in which the ionic configurations of importance are the molecular, CI+C1 - , ion-pair states. Three principal starting points motivate this study. From the viewpoint of identifying many-body contributions to interaction potentials, as they arise in condensed phase systems, we are interested in establishing a hierarchy of approximations based on sound physical principles and transparent insights. Different approaches, therefore different languages, can be used for such formulations. The energy decomposition analysis of Morokuma [1], the natural bond orbital analysis of Weinhold and coworkers [2], and perturbation theory-based formalisms [3-8], are examples. We have chosen the DIM framework for our approach, and consider the extent to which 'non-additive' contributions to many-body interactions can be reproduced with information solely derived from a knowledge of the constituent
0301-0104/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. P 1 1 S 0 3 0 1 - 0 1 0 4 ( 9 7 ) 0 0 1 0 7 - 9
162 B.L. Grigorenko et al. / Chemical Physics 219 (1997) 161-
pair potentials. Given the fact that information about diatomic fragments can be derived with high levels of accuracy, properly devised schemes for using such information in treating extended systems is of general and practical value. An analysis of the HeC12 potential energy surface both in the ground (X) and excited (B) electronic states of the molecule was recently reported [9]. There, the DIM method was applied with a complete set of polyatomic basis functions arising from the lowest energy states of the atoms CI(2p) and He(1S). Several levels of the theory, based on inclusion of various contributions to the interaction potential were tested. It was discovered that the greatest impact on the shape of the potential energy surface, in particular in the regions of the potential wells, stems from the very first step in the procedure, when the pairwise potential is extended to include the anisotropy of the He-Cl interaction. Other refinements within this limited basis lead to less noticeable changes. In the present work, we consider the effect of augmenting the DIM basis set to include ion-pair states while at the same time reducing the matrix of neutral states to a bare minimum. We show, that the inclusion of a single ion-pair configuration, together with the anisotropy of covalent interactions, leads to a significant improvement of the three-body surfaces. The importance of ion-pair states in the DIM description of polyatomics has been highlighted before. In previous DIM applications it has been shown that the inclusion of ionic states is essential for the correct prediction of equilibrium geometries in molecules of partly ionic bonding character [ 10,1 l ]. The same has been clearly emphasized in the diatomics-in-ionic-systems (DIIS) treatment of halogens in rare gas solids by Last and George, and coworkers [12-14]. Our recent calculations on van der Waals clusters of HFAr n [15], and (HF) 2 Ar n [16], demonstrated the utility of DIM-based A r - H F three-body surfaces which were constructed by inclusion of ionic states. The crucial contribution of the latter to produce an acceptable three-body ground surface was detailed through comparisons with the accurate ab initio and empirical potentials for the same system. While the strategy of including ion-pair states in the DIM matrix has been demonstrated to be useful in constructing three-body surfaces, it must be appreciated that they will have a more profound effect in condensed phase treatments. Due to the long-range nature of ionic potentials, they are subject to truly many-body solvation in a fundamentally different way than neutrals which are dominated by short range interactions. The admixture of ionic configurations in ground state electronic surfaces can therefore be expected to have dramatic effects in the non-additivity of potentials constructed from neutral fragments alone. Finally, the structure of vdW complexes between rare gas atoms and molecular halogens, Rg-X 2 (where X is a halogen atom), has been the subject of extensive deliberation. In the electronic ground state, two competing geometries of these complexes are at issue: T-shaped vs. linear. Given the very weak binding in these complexes, it is not too surprising that the precise distinction between the two possible isomers is a serious challenge to theory. According to most recent high level ab initio calculations the potential minimum for the linear R g - X - X configuration is = 5-7% lower than that of the T-shaped structure [3,4,7,8]. Despite this, due to the consideration of zero-point energies, the ground state wavefunction in He-Cl2 is localized in the T-well, consistent with experiment [3,4]. At the same level of theory, the same type of compensation does not occur in the case of the heavier Ar-C12 complex, where the predicted minimum energy structure is linear, in apparent contradiction with experiment [3,4]. At much higher levels of theory, the linear well while deeper, approaches the T-minimum, to now favor the T-structure when the zero-point effect comes into play [4]. Quite clearly, the use of isotropic atom-atom potentials can only yield the T-shaped isomer for the van der Waals complex. The additional linear minima can only arise when anisotropic R g - X pair potentials are considered, as illustrated by Naumkin and Knowles for A r - X 2 surfaces [17]. These treatments can be regarded as the lowest in the hierarchy of DIM constructs, as already elaborated [9]. However, even with the use of the more complete treatment, including all 36 molecular halogen states in the DIM basis, in the case of He-C12, the minimum for the linear geometry is essentially absent [9]. As surmised, the incorporation of ionic configurations in the DIM basis changes this situation. In this work, we first consider the effect of ionic contributions on the singlet ground state surface of Rg-CI 2 by taking the minimal basis set required for such a treatment. We include in the DIM set of polyatomic basis functions the lowest energy ion-pair states which mix with the ground state, namely, states derived from the
164 B.L. Grigorenko et aL / Chemical Physics 219 (1997) 161-
The fragment Hamiltonian matrices are given as:
H(CZ:) = Bc]12V(C12)Bcl 2 [-I,C"~'Rg) = [RCl~,] *7(Cl(i)Rg)BCl,~,, , ( 2 ) H (Rg) = V(Rg) i.l(c~"') = 7(C1(O)
where all 7 matrices are diagonal, containing energies of the diatomic and atomic states (Table 1). The rotation matrices R, rotate the quantization axis from the Rg-C1 direction to the reference frame of the molecular halogen, specifically, for i = l:
cos 4~1 sin ~bI
- s i n ~ b I cos4~l COS ~
- sin &l 9 % =
sin 4h COS (~ ½(3cos 2 Oh,- 1) - ~ - c o s ~ b I sin0b, -}!/~sin24~, cos ~b1 sin ~b1 cos2 ~bl - sine 4q - sin ~bI cos 4~j ½!/~-sin2qS, sin ~, cos 4~, ½(1 + c0s24), ) 1 1 1 (3)
The matrix B is responsible for mixing the diatomic states of the same spin and spatial symmetry, here, the mixing is between the three ] £ states of CI 2 (see Table 1 (a)). Most generally, the 3 × 3 non-unit block of the orthogonal matrix B contains three mixing parameters a, /3, 3":
F cos c~ cos 3, - cos 13 sin ~ sin 3" cos a sin 3" + cos/3 cos 3' sin a sin ce sin/3 ] B -- J - sin o~ cos 3' - cos /3 cos ~ sin 3' - sin c~ sin 3' + cos /3 cos 3' cos a c o s a s i n / 3 ]. (4)
/ /
[. sin/3 sin 3' - sin/3 cos 3' cos/3 _]
We simplify the problem by setting a = 3' = ~r/4, and by estimating /3 from electronic structure calculations. To this end, we created a set of molecular orbitals for the C12 molecule at its equilibrium position, then
1 ~ ~ x Fig. 1. Coordinate system used for the R g - C l 2 complex.
B.L. Grigorenko et al. / Chemical Physics 219 (1997) 161-172 165
transform these orbitals to a set of natural atomic orbitals, namely, the localized atom centered one electron functions [2]. With this set of orbitals we performed the CI calculation when distributing all valence electrons over all valence orbitals of the molecule. Analyzing the multiconfigurational wavefunctions we concluded that the weight of the covalent contribution in the ground state of CI 2 is approximately 0.8, and the ionic contribution is about 0.2, i.e. /3 = 25 °. Finally, as already discussed in our previous work on the A r - H F DIM potential energy surface, the potential matrix elements for the ion-pair states should be summed vectorially, as the sum of ion-atom interactions [15]. Restricting this interaction to the leading electrostatic term of charge-induced dipole, it is convenient to use scalar pair potentials for the ion-atom interactions and to correct for the combined action of two oppositely charged centers, here C1- and C1 +, with the correction term given as:
Agio n = c~(Rg) 3 3 (5) RRgCI~ ~)RRgCI~2~
3. Diatomic input
For the Rg-CI interactions we use the experimentally determined adiabatic potential curves for the ~ and H states which were derived by Aquilanti et al. from measurements of total scattering cross-sections [20]. We use their fits for the 2]~ and 2[I states of He-C1, Ar-C1 in the form of the MSV (Morse-Spline-van der Waals) parametrization for isotropic part of the interaction and the Buckingham form for the anisotropic component [20]. For He-CI (Is), and A r - C I - ( I s ) potentials we use the extended Tang-Toennies (ETT) potentials which are based on a combination of SCF calculations at short range, and scaled induction and dispersion terms up to R-l0 for long range interactions [21]. For the Rg-CI + system, we are only aware of the paper by Balasubramanian et al. on Ar-CI + [22]. They use the relativistic pseudopotential developed by Pacios and Christiansen [23], and use the corresponding [3s3pld] basis sets within the first order configuration interaction (FOCI) scheme followed by the valence shell CASSCF procedure. We recalculated the i~, l ii ' 1A states of Ar-CI+(ID), along with those of H e - C I + ( I D ) by using an advanced procedure. Namely, the diatomic energies are computed with the help of second order CI (SOCI), taking into account single and double excitations from the set of reference configurations which in turn account for complete distribution of valence electrons over valence orbitals. The calculations were carried out with the
Table 2 Coefficients of the fitting function, Eq. (6) in text, for the C I + - R g potentials. Distances are in a.u., energies are in atomic units A (au) a ( l / a u ) B ( l / a u ) C ( l / a u 2 ) D ( 1 / a u 3) Limits HeC1 +
ArC1 +
~Z 3. 9 2 4 × 10 - 2 1.356 8.5 - 2. 2 1 - 0. 0 8 r > 2. I H 0.2528 1.680 16.59 - 2. 0 7 - 0. 2 2 r > 2.5 ,~ IA 0.4873 1.712 19.29 -- 1.90 --0.21 r > 2.5 ,~ I ~ 9. 9 7 6 × 10 - 3 0.863 --44.72 --8.21 1.78 r > 2.5 A, LH 3,641 1,178 4.080 - 1.58 0.14 r > 2. IA 0.2874 1.768 168.4 - 5. 3 6 0.04 r > 2.5 ,~ 3 II 2.022 1.240 5.20 - 2.02 0.18 r > 2.7 ,~
B.L. Grigorenko et al. / Chemical Physics 219 (1997) 161-172 167
4. Results
4.1. The ground state
We compare the shapes of the HeCl 2 and ArCI 2 interaction potentials for the fixed CI-C1 internuclear distance of 1,988 A and regions of distances from Rg to the center of mass of CI 2 up to 5 ]~, for three treatments: (A) isotropic pairwise additive atom-atom potentials, in which for the Rg-C1 interaction we use the ground state 2E potential:
V= vC'C~+ Y'. vRgC~(Ri). (7)
i = 1. 2 (B) DIM surface limited to only neutral configurations, only the first four CI 2 states in Table 1, in which the anisotropic Rg-C1 potentials are included. This is obtained as the lowest root of the 4 × 4 secular equation. Note, if only the diagonal matrix element H(4,4) of Table 1 is taken into account without any coupling to other states, then we would arrive at the Naumkin-Knowles (NK) model, where [17]:
V~ov= E [v,~gc'( Ri) sin2 &i + v~gC'( Ri) c°s2 d>i]. (8)
i - 1. 2 (C) DIM surfaces with the inclusion of ion-pair states, obtained as the lowest root of the secular equation of the 10 × 10 matrix described in Eq. (1). The resulting potential energy surface can be formulated, to explicitly show the various contributions, through a procedure equivalent to what leads to the approximation of Eq. (8). This is accomplished by converting the 10 × 10 Hamiltonian matrix form the initial polyatomic basis set shown in Table 1, with mixings, to the set that diagonalizes the CI 2 fragment Hamiltonian (with corresponding transformation of the Rg-CI i matrices). Then, the neglect of couplings among the lowest energy matrix elements to all other states, leads to the following expression:
V = Vco v c o s 2 f l -~ Vio n sin 2 fl, (9) where V~ov is given by Eq. (8), while:
g i o n = A V i o n -I- ~ (R,) + , i 2 with AV~o. defined in Eq. (5), a single potential describes the closed shell-closed shell Rg-C1- interaction (given in Table 3); while the closed shell-open shell interaction with the positive ion, Rg-CI +, is given as:
3 sin 4 ~bi vRgCl+(RI) = (3 cos e 4,i - 1) 2 vR~CI+(Ri) + 3( sin2 'hi cos" 4' ~V Rgc'+ ~ Ri) + - - 4 ij H t 4
vR CI+ ( Ri) ,
(11) with parameters given in Table 3. Fig. 2 shows contour plots of the ground state potential energy surfaces of HeC12 for the three treatments. The C12 axis is placed along the horizontal, and the energy contours are drawn in 2 cm-~ increments starting from the surface minimum. Table 4 contains parameters of the stationary points of these surfaces. The comparison allows an assessment of the different contributions to the surface topology. The inclusion of the anisotropy of the Rg-C1 interaction (a vs. b) leads to the appearance of the linear well, however, this well is separated from the T-valley by an insignificant saddle point of less than 2 cm-~ in height. Inclusion of the ion-pair contributions (b vs. c) brings the wells closer in energy, by reducing the T-well minimum, while creating a well defined saddle point between the two minima. The main conclusion from this study is that the T-shaped and linear configurations of HeCI 2 become more competitive as one climbs from models A through C.
168 B.L Grigorenko et aL / Chemical Physics 219 (1997) 161-
The calculation results for ArC12 are presented in Fig. 3 and in Table 4, in the same format used for HeC12. In this case the assessment of the effect of the ionic configuration is more obvious. Now, the DIM scheme with inclusion of ion-pair states predicts that the linear form of ArC12 is lower in energy. As before, the linear minimum already arises from the anisotropy of the Ar-C1 interaction (case B), although significantly less deep
a ) i i ~ i i i I i i
4.00j
3.00- ~ ,
2.00- ~
1,00-
0.00 (^) ] I I I I / I I -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.
(b) s oo/ C^ ' ' L~:' ' i ' '~--~ ' J~--
3.00-t
2.00-
1.00-
0.00 ~ , ~ , ~. f -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5,
()c /! _ '
i.ioo
-5.00 -4.00 -3.00 -2.00 -1:00 0.00 1.00 2.00 3.00 4.00 5. Fig. 2. Contour plots HeC12 , plotted at 2 c m - i resolution. Top panel: isotropic atom-atom potentials, scheme A of text; middle panel: DIM surface with only neutral CI 2 states, scheme B of text; bottom panel: DIM surface including ion-pair state contribution, scheme C of text.
170 B.L. Grigorenko et al. / Chemical Physics 219 (1997) 161-
4.2. The excited state
Within the same approximation as above, it is possible to give an analytical expression for the excited state, CI2(3II)-Rg, potential energy surface. Starting with the 6 × 6 matrix in Table 1 (b), therefore limited to the single C1 ÷ C1-(3II) ionic contribution. As before, the overall surface is given by the admixture of covalent and ionic contributions, of Eq. (9), except now the covalent contribution is isotropic, the average of ~ and 17 interactions of Rg-X:
Vcov= ½ E [vRgCl(Ri) + vffgCl] , (12) i=1,
while the ionic contribution is given as:
Vio.=AVio. +½[ E [V'I]C'+ ( Ri) + vR~CI-( Ri)] • (13) i=1,
where the RgCI + terms limited to the 3II curve. The resulting surface is shown in Fig. 4 for the case of CI2(3II)-Ar. Note, in this case, the inclusion of ionic configurations does not produce linear minima. The T-minimum in this case is at an internuclear distance of 3.6 A, with a dissociation energy of 216 cm -.
4.3. The DIM insight
An attractive aspect of the DIM construct is that it provides simple insights about the final surfaces. To this end, we consider the hierarchy of effects. First, let us consider neutral interactions alone. An R g - X 2 surface based on isotropic R g - X potentials would only have a T-shaped minimum, in which the Rg atom would have the benefit of stabilization by two X atoms via R - 6 dispersion. The anisotropy of the Rg-X interaction results from the repulsive Pauli exclusion force. In the ~ state, where the halogen hole points at the Rg atom, a closer approach, and therefore deeper binding via dispersion is possible [20]. Since the Rg-X interaction is much weaker than the covalent bond in the molecular halogen, the quantization axis obtained by diagonalization of the DIM matrix remains essentially unaffected by Rg. Accordingly, in the 1~ ground state, the Pz hole on each of the halogens implies dimples in the electron density for linear approach of the rare gas. This is the essence of Vcov in Eq. (8). Whether linear minima will result, depends on whether the anisotropy of the R g - X potentials, i.e., the difference between ~ and 17 approaches, is large enough to overcome the double coordination in the T-geometry. This does not occur in He-C12 and Ar-C12. In the excited 31-I state of X 2, the hole on each X atom has an equal contribution from Px and Pz, hence the R g - X interaction is isotropic, the average of II and ~ potentials, as in Eq. (12). Accordingly, the minimum in the three-body excited surface would be exclusively limited to the T-geometry. The inclusion of ionic configurations incorporates weighted contributions from the atom-ion pair potentials. We need only focus on the anisotropic R g - X ÷ fragment, in which the interactions are a balance of electrostatic
4.0~.
2.0 i
0. 0 - -6.0 -4.0 -2.0 0.0 2.0 4.0 6. Fig. 4. Contour plots ArC12(X 311) plotted with 20 cm- 1 resolution.
B.L. Grigorenko et al. / Chemical Physics 219 (1997) 161-172 171
attraction and Pauli repulsion. Since Rg represents a spherical charge distribution, the anisotropy is strictly the result of the repulsion. Thus, the dramatic difference among 1~, 11i , 1A potentials of Ar-CI+(ID), see Table 2, is the result of Ar approaching a doubly unoccupied, singly occupied, or doubly occupied orbital, respectively. The same consideration explains the similarity between l[I and 31-I states, and 1A and 3~ states [22]. It is the contribution of the R g - C I + ( I ~ ) potential, which is bound by nearly 1.5 eV, that leads to the generation of the linear minimum in the singlet ground state. Also, the absence of binding in RgCI+(3]~) while RgCI+(3H) is bound by = 0.3 eV, simply reinforces the T-shaped minimum on the triplet surface when the ionic admixture is added.
5. Discussion and conclusions
The general features of the ground state potential energy surfaces of He-C12 and Ar-C12 obtained by the inclusion of ionic configurations in the DIM basis seem realistic in that they show similar stabilities for the linear and T-geometries of the complex, a feature that is absent when only neutral interactions are considered in the most thorough treatment [17]. The ion-pair states give the effect sought on the topology of the surface, and account for the recognized non-additivity of pair interactions. In comparison to the best ab initio surfaces, the deviations in the constructed surfaces range from = 10-50 cm i for the various stationary points that are compared in Table 4. There is no comparable ab initio data for the excited state. Spectroscopically, it is known that the T-minimum in the B(31-Iu) state is = 12 cm 1 shallower than in the ground state [29]. Using the converged values between experiment and highest level of theory for the ground state [4], we can back track a binding energy of = 200 cm- 1 for the excited state minimum, to be compared with 216 cm- l in the present (a 5% deviation). Since the DIM predictions are critically dependent on the quality of the diatomic input, an assessment of their accuracy is useful for putting the observed deviations in perspective. The most accurate of the input potentials used are the Rg-C1 adiabatic curves for ~ and 11 states, which are derived by Aquilanti et al. from experiments, and which are significantly more accurate than the observed deviations [20]. The closed shell Rg-CI-- potentials are also deemed reliable. Nevertheless, the He-C potential used here has been criticized by Diercksen and Sadlej [27]. They noted that the long-range part of this potential was based on outdated and under estimated polarizabilities for CI-. However, due to cancellation of errors, the final parameters for the H e - C I - interaction were considered satisfactory. Relying on the careful analysis of H e - F and H e - C I - by Dierksen and Sadlej [26,27], we estimate possible errors of order 10- cm t in the range of interest to our calculations. The Rg-C1 + potentials are subject to the largest errors. The singlet states correlating with the Ar + CI+(ID) dissociation limit were previously computed using first order CI [22]. We repeated these calculations by dramatically extending the configuration space. The differences in interaction energies in the two procedures was = 20%, which corresponds to uncertainties of several hundred wavenumbers in the range of interest in the present. Thus, the deviations of the devised surfaces from the best ab initio results are well within the uncertainties of the input parameters in our construct. Despite the uncertainties in diatomic input energies, the qualitative conclusions of the DIM predictions remain firm. Even assuming severe modifications in fragment potentials the energies of linear and T-shaped forms of HeC12 become competitive upon inclusion of ion-pair contributions. What is affected by the possible modifications of the energies of diatomic fragments are the precise values of binding energies of HeC12, within 10 cm - ~. For ground state ArC12 the conclusion of the greater stability of the linear form in the DIM treatment we deem to be outside the possible errors in the diatomic input. Similarly, the absence of the linear minima in the excited state is a reliable conclusion. These features are absent in DIM surfaces, such as the recent work of Buchachenko and Stepanov [30], in which ionic configurations are ignored. Quite clearly the details of the surface topology can be modified by a suitable adjustment of input parameters, or alternatively, the functional form of Eq. (9) can be used as a fitting scheme to generate surfaces for applications requiring high accuracy. What concerns condensed phase applications, where many-body contributions control structure and dynam- ics, it is important to have surfaces that have properly parametrized non-additivity. The present approach seems
