Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 140 (2015) 305–310
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A refined quartic potential energy surface and large scale vibrational calculations for S0 thiophosgene Svetoslav Rashev a,⇑, David C. Moule b a b
Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradskochaussee 72, 1784 Sofia, Bulgaria Department of Chemistry, Brock University, St. Catharines, ON L2S3A1, Canada
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
We compute an ab initio quartic force
field for S0 thiophosgene Cl2CS. We calculate vibrational energy levels
using our specific variational method. We carry out an adjustment
procedure on the ab initio computed quartic PES. We calculated a large number of excited vibrational level energies. We obtained a good fit between calculated and experimental frequencies.
a r t i c l e
i n f o
Article history: Received 13 October 2014 Received in revised form 8 December 2014 Accepted 28 December 2014 Available online 6 January 2015 Keywords: Thiophosgene Potential energy surface Refined surface Variational vibrational calculations Experimental frequencies
a b s t r a c t In this work we present a full 6D quartic potential energy surface (PES) for S0 thiophosgene in curvilinear symmetrized bond-angle coordinates. The PES was refined starting from an ab initio field derived from acc-pVTZ basis set with CCSD(T) corrections for electron correlation. In the present calculations we used our variational method that was recently tested on formaldehyde and some of its isotopomers, along with additional improvements. The lower experimentally known vibrational levels for 35Cl2CS were reproduced quite well in the calculations, which can be regarded as a test for the feasibility of the obtained quartic PES. Ó 2015 Elsevier B.V. All rights reserved.
Introduction Thiophosgene is an interesting molecule, that has attracted considerable attention both experimentally as well as theoretically, as it has proved to be a very suitable model system for studying of a large variety of generic spectral and photophysical manifestations in polyatomic molecules [1–12]. Thiophosgene has been recognized both as a suitable molecule for studying ‘‘backbone’’ intramolecular vibrational energy redistribution (IVR) in the ground ⇑ Corresponding author. E-mail address: [email protected] (S. Rashev). http://dx.doi.org/10.1016/j.saa.2014.12.101 1386-1425/Ó 2015 Elsevier B.V. All rights reserved.
electronic state S0 [3,4,8,9], as well as ‘‘a molecule, tailor-made for studying fundamental concepts of electronic radiationless transitions’’ [1,2,5–7]. The vibrational spectroscopy of thiophosgene has also been studied in considerable detail, although even at the present time some vibrational fundamentals have not been directly observed. As a result of an extensive research effort, many ground electronic state frequencies have been measured up to extremely high vibrational excitation energies, of 23,000 cm1, using low resolution IR [13–16] and high resolution synchrotron IR spectroscopy [17], laser induced fluorescence and stimulated emission pumping from both excited electronic states S1 and S2 [4,8,11,19,20]. This range of observed vibrational frequencies
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extends up to and even exceeds the first two dissociation limits (19,900 cm1 to CS + Cl2, and 22,200 cm1 to Cl + SCCl). A surprisingly large number of the spectroscopically observed highly excited vibrational levels (several hundred), including many levels above dissociation limit, have been successfully observed and assigned using a simple nonresonant effective Hamiltonian expression by Gruebele and collaborators with only 20 adjustable parameters [4,11]. This suggests, that characteristic vibrational feature states – well isolated quantum states with assignable quantum number composition – should exist in the whole range of S0 vibrational excitation and even above the lowest dissociation limits [11]. Many theoretical models, both quantum mechanical and semi-classical, have been designed and applied to the understanding of the extent and mechanisms of vibrational level mixing and vibrational energy redistribution (IVR) in S0 thiophosgene [4,9– 12]. In particular, rationalizing the existence of numerous highly excited assignable vibrational states, not strongly affected by vibrational mixing – which contradicts simple statistical theories – is a major theoretical challenge. The existence of such states suggests the possibility for controlled monomolecular reactions, through the selective initial excitation of appropriate vibrational levels. Several model 6D potential fields for S0 thiophosgene have been calculated so far in the literature [4,21,22]. The ab initio field of Strickler and Gruebele [4] has been calculated with bond distance coordinates and then refined using over 100 vibrational levels up to 8000 cm1. The recent quartic field by Davisson et al. [22] was computed using a CCST(T) method, in terms of symmetrized internal coordinates, and then transformed into Cartesian coordinates. The authors’ objective in this work was to show, that it was highly advantageous computationally to use hybrid basis sets in the quantum mechanical computations, namely an aug-cc-pVQZ for the equilibrium geometry and harmonic force field, while a ccpVTZ for the cubic and quartic force constants. The hybrid approach albeit highly economic, was shown to be of only slightly inferior quality as compared to the use of an aug-cc-pVQZ throughout. For their vibrational calculations, Davisson et al. [22] used the so called VPT2+K approach, which is based on second order vibrational perturbation theory with resonances treated explicitly using a Hamiltonian formalism. Our earlier empirical field [21], was of limited scope and applicability since our vibrational calculation method involved a simplified (approximate) expression for the kinetic energy operator and in addition the field included only the quadratic and several (small part) of the cubic and quartic force constants. In this work we present a full quartic PES for S0 thiophosgene (including all 34 quadratic, cubic and quartic force constants), in terms of the symmetrized curvilinear coordinates that were used by MLT [23] and Handy [24] in their work on formaldehyde (different from the coordinates used in our previous work [21] and from those used by Strickler and Gruebele [4]). The field obtained in this work appears to be of good quality, as it was able to reproduce the experimentally measured lower excited vibrational levels (fundamentals, overtones and combinations) in S0 thiophosgene. Using this field and our recently developed variational method, we carry out detailed large scale calculations on the vibrational level structure of S0 thiophosgene up to 3000 cm1. We calculate vibrational levels of all symmetries and compare our results to the experimentally observed frequencies in this energy range. In most cases we provide our own assignments and compare them to the assignments by other authors. We also discuss the most important resonance interactions (of Fermi and Darling-Dennison type). In forthcoming work, using the obtained PES, it will be our purpose to explore the vibrational mixing patterns in S0 thiophosgene at very high vibrational excitation and perhaps check the validity of the restricted vibrational models, that have been employed by other authors for this purpose.
Vibrational Hamiltonian and vibrational basis set We employ our Hamiltonian formalism, Hvib = T + V, that we recently developed and described in detail for the exploration of the vibrational level structure and vibrational level mixing in S0 formaldehyde and isotopomers [25–28]. The kinetic energy expression T is the same exact expression (with changed nuclear masses) as used for formaldehyde (Handy [24]), while the quartic PES V is described as a quartic Taylor expansion [23],
V ¼ E0 þ
X X X F ij Q i Q j þ F ijk Q i Q j Q k þ F ijkl Q i Q j Q k Q l ; ij
ijk
ð1Þ
ijkl
In terms of 6 symmetrized curvilinear coordinates Qi, i = 1, . . ., 6, defined as follows (C2v symmetry species are indicated): p p Q1(A1) = qC–S, Q2(A1) = (q1 + q2)/ 2, Q3(A1) = (h1 + h2)/ 2, Q4(B1) = p p sCl–C–S–Cl0 , Q5(B2) = (q1 q2)/ 2, Q6(B2) = (h1 h2)/ 2. Here q1,2 = Dr1,2/(Dr1,2 + r0), qC–S = DrC–S/(DrC-S + r0,C–S) are three Simmons–Parr–Finlan (SPF) coordinates [29] for the two stretches Dr1,2 from equilibrium r0 of the two Cl–C bonds, and the stretch DrC–S from equilibrium r0,C–S of the C–S bond, respectively; h1 and h2 are the two bends from equilibrium h0,Cl–C–S of the two Cl–C–S angles and sCl–C–S–Cl0 is the change of the ‘‘book’’ angle between planes Cl1–C–S and Cl2–C–S, from equilibrium p. Both kinetic energy [24] and potential energy (1) expressions are presented in separable form (sum of products of 1D functions), therefore the 6D basis wavefunctions that form the primitive basis set, can conveniently be used in the form of products of 1D basis functions corresponding to each of the local vibrational coordinates: q1,2, qC–S, h1,2, sCl–C–S–Cl0 . In the following we shall briefly specify our definition and construction of the 1D basis sets for each of the curvilinear coordinates. The basis functions for each of the three stretches Cl1–C, Cl2–C and C–S respectively, are originally taken as 31 Morse oscillator eigenfunctions, |n1i, |n2i, |n3i, n1, n2, n3 = 0, . . ., 30. With a suitable choice of the Morse parameters (harmonic frequency and anharmonicity corresponding to the type of molecular vibration described), these sets of basis functions can be adapted to give a distribution of energy levels En = hn|Hvib|ni that roughly corresponds to the relevant molecular fundamental and overtone vibrational frequencies. This is the type of basis that we have employed in our recent calculations on formaldehyde and isotopomers [25–28]. However these basis functions have large and increasing with nnondiagonal matrix elements hn|Hvib|n+1i, which exceed the energy separation of adjacent levels En+1 En, at the larger n values. This is not convenient for the operation of our search selection algorithm, especially when exploring very highly excited overtone excitations. Therefore we performed optimization on the Morse functions, by prediagonalization with a simple appropriate 1D Hamiltonian, which yields a new set of basis wavefunctions, that are free from the above mentioned defect. These basis functions will be denoted from now on as |n1i, |n2i, |n3i, n1, n2, n3 = 0, . . ., 30, for each the stretches Cl1–C, Cl2–C and C–S, respectively (31 basis functions of each type). For the out of plane bend, we employ 38 harmonic oscillator eigenfunctions |n4i, without additional prediagonalization. Finally, for the two S–C–Cl(h) bends we start with a set of normalized associated Legendre polynomials P 2n ðcos hÞ, n = 2, 3, . . . that cancel the singularities in the KE operator T. These wavefunctions have no free parameters to adjust and are not well adapted to the molecular vibrational levels, therefore for them we also apply a prediagonalization of the 1D basis using an appropriate simple Hamiltonian, as described in detail recently [26]. The basis functions that are obtained from this procedure, are denoted as |n5i, |n6i (n5,n6 = 0, . . ., 48), for the two bending modes h1,2, respectively. As a result of the separable form of the Hamiltonian, a 6D basis function can be presented in product form: |n1, n2, n3, n4, n5, n6i = |n1i |n2i |n3i |n4i |n5i |n6i and a Hamiltonian
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matrix element hn1, n2, n3, n4, n5, n6|Hvib|n01 , n02 , n03 , n04 , n05 , n06 i is reduced to a sum of products of factors, that are obtained through 1D numerical integration. The dimensionality of the primitive basis Q set is N p ¼ 6i¼1 n0i ¼ 1; 608; 161; 632. To accelerate the large scale calculations, prior to each actual vibrational calculation, we i;a compute a number of 2D arrays Pni ;ni 0 ¼ hni ðqi ÞjF ai ðqi Þjn0i ðqi Þi, i 0 ni, ni = 0, 1, 2, . . ., n0,i, for each vibrational coordinate qi and each function or operator depending on this coordinate F ai ðqi Þ, occurring in either KE or PES expressions, using either Gauss–Hermite, Gauss–Laguerre or Gauss–Legendre numerical integration [30], where n0,i is the number of basis functions employed for each vibrational coordinate qi. All computed n0,i n0,i0 arrays are stored in computer core memory, ready to use in the subsequent matrix elements calculations. Now, each matrix element can be obtained ai as the sum of products of the appropriate P i;m;n values hence reducing the actual calculation to a series of multiplications and summations and no integrations, which greatly accelerates the calculation of matrix elements. In our calculations we employ symmetrized basis functions |n1, n2, n3, n4, n5, n6; Si, that are obtained as linear combinations of the original product wavefunctions |n1, n2, n3, n4, n5, n6i. E.g., if in a product state one or both pairs of degenerate modes have different quantum numbers (n1 – n2 and/or n5 – n6), we symmetrize by taking |n1, n2, n3, n4, n5, n6; Si = (|n1, n2, n3, n4, n5, n6i ± |n2, n1, n3, n4, n6, n5p 3, n4, n5, n6; Si = (|n1, n2, n3, n4, n5, n6i ± |n2, n1, n3, n4, n6, n5i)/ 2, where symmetry can be S = A1,B2, if n4 – even or S = B1,A2, if n4 – uneven. Thus, our basis functions are not the usual normal modes, but they are superpositions of symmetrized local modes excitations.
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the selected AS (for increasing R and decreasing C values), which will result in enhanced accuracy and display the convergence in the calculation of the desired molecular vibrational levels. All selected basis states are stored as an array in computer core memory. Simultaneously the Hamiltonian matrix is built that contains the diagonal and nondiagonal Hamiltonian matrix elements of the selected basis states. The Hamiltonian matrix H constructed in the course of the search/selection procedure, besides being optimal in size, is also quite sparse, because the algorithm employed automatically discards the matrix elements that are too small according to the criteria of the search. This makes our vibrational variational procedure both memory and time efficient. For the tridiagonalization of H we employ a conventional Lanczos iteration without reorthogonalization [31,32], starting with the vector |0i. We diagonalize the resulting tridiagonal Lanczos matrix using the routine tqli() from numerical recipes [30], in slightly modified form. From this calculation we obtain the spectral distribution |Ci|2 in the explored range of vibrational excitations, where Ci is the first component (corresponding to basis state |0i) of the eigenvector |ii, at eigenenergy Ei. The dilution factor r [33] is a measure of the vibrational fragP mentation around |0i and is given by the expression: r ¼ k jC k j4 [9]. In fact, Neff = r1 accounts for the average number of vibrational eigenstates |ii that are effectively intermixed with (coupled to) the feature (or ‘‘bright’’) state |0i i.e., it may be considered to represent the extent of vibrational mixing around |0i.
Variational vibrational calculation procedure for thiophosgene
Calculation of an ab initio quartic PES for thiophosgene and adjustment of the ab initio calculated PES to the experimentally measured frequencies
Our specific search/selection procedure [25] serves to select an optimally small however representative active space (AS) of NAS basis states (from a huge available primitive space Np), that are most relevant to the particular vibrational calculation and to simultaneously construct the relevant Hamiltonian matrix H. A search/selection procedure is started from a particular basis (feature) state |0i, chosen to be the best zeroth-order representation of the vibrational levels that we are interested in. The algorithm is symmetrically adapted to search and select only such basis states whose symmetry coincides with the symmetry species of the initial state |0i. During the implementation of the search/selection algorithm, many basis states are being probed and each state that satisfies the criteria for sufficient coupling strength is selected and consecutively added to the previously selected AS. Although the employed optimized search/selection algorithm is in itself a rather complicated sequence of procedures, there are essentially two major parameters, determining the scope and the quality of the search, C and R (whose values have to be fixed at the outset), that have been defined and discussed in our previous work [25]. R (a filtering parameter, usually chosen as 1000 cm1) is a window around the zeroth energy of the initial (for the search) basis state |0i, which acts to promote the selection of basis states that are located within the energetic R vicinity of |0i and to prevent the selection of too strongly displaced basis states. An evaluation function value fk is ascribed to each probed for selection basis state |ki, accounting for its (both direct and indirect) coupling strength to |0i. According to its definition [25], fk is 1 for the initial state |0i and attains smaller and diminishing (but positive) values for all subsequently selected states (R is also involved in the determination of fk). C (usually chosen as 1010 or smaller) is the limiting value for fk. Probed basis states with fk < C will not be selected. According to the values chosen for the parameters C and R, the search/selection procedure will select a varying number of basis states, i.e., include more and more weakly coupled basis states into
First, a full quartic potential field for thiophosgene was generated with the nwchem suite of ab initio programs [34]. The approximation used for this PES study was the cc-p VTZ correlation consistent polarized triple valence basis. This was further corrected for electron correlation by CCSD(T). This is a coupled cluster method with single and double substitutions augmented by a perturbative estimate of the triple excitations. To create the grid of structural points as input for the energy calculations, the bond lengths were adjusted in step sizes of 0.05, 0.1, 0.2 Å, while the bond angles were varied by increments of 5° and 10°. The potential energy function was fitted to 1083 resulting data points with the nonlinear regression package NLREG [35] with a mean square error of 36 cm1. In Table 1 we present the results for the 6 fundamental 35 Cl2CS frequencies, calculated from the set of ab initio computed force constants in this work (column 3) and compared to the experimental data (column 2) and those calculated by Davisson et al. [22], using their ab initio computed hybrid basis set (column 4). From a comparison of the vibrational data, obtained using both ab initio fields with the experimental values it can be seen, that our ab initio quartic force field is somewhat inferior to that of Davisson et al. [22]. Next, the ab initio calculated quadratic, cubic and quartic force constants were refined, using our large scale variational method and a nonlinear least squares fitting algorithm (Marquardt’s method for nonlinear parameter estimation through the ‘‘chisquare’’ minimization, implemented in the routine mrqmin() from [30]. Our refinement procedure was similar to that described in our recent work for formaldehyde [28]. In addition to the force constants, the three equilibrium parameters (two bond lengths and one interbond angle) were also varied. We compared the calculated vibrational frequencies, to a selected set of 19 experimentally measured values in the lower excitation range, up to 3000 cm1. We carried out careful analysis of the convergence by varying the parameters of the search (decreasing the value of
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Table 1 Fundamental 35Cl2CS frequencies, calculated from the set of ab initio computed force constants in this work (column 3) and compared to the experimental data (column 2) and those calculated by Davisson et al. [22], using their ab initio computed hybrid basis set (column 4); our frequencies calculated from the refined PES, are given in column 5; the experimental frequencies (column 2) were taken from (a) Ref. [11], (b) Ref. [16], (c) Ref. [17], (d) Ref. [18]. Fundamental frequency
m1 (A1) m2 (A1) m3 (A1) m4 (B1) m5 (B2) m6 (B2)
Experiment
1139.70(a) 503.80679(c) 292.80(a) 471.04267(c) 819.614(d) 305(b)
Calculated ab initio (this work)
Calculated ab initio in [22] (hybrid basis set)
Calculated from the refined PES
1134.92 500.03 303.50 469.85 812.87 309.11
1138.5 502.7 292.5 469.8 820.6 301.8
1138.58 503.38 293.61 470.42 814.97 305.20
C and increasing the value of R led to enhanced accuracy of the calculated frequencies, but to increased calculation times, as well). It was found, that a calculation with C = 1012 and R = 1500 cm1 is sufficient to achieve an accuracy of 0.05 cm1 for the lower excited thiophosgene frequencies in the range 0–3000 cm1. This resulted in selected active spaces of about NAS = 30,000–50,000 basis states. In our least squares fitting procedure we varied the quadratic force constants first, before turning to the higher order constants. Although thiophosgene has been the object of numerous spectroscopic studies by many authors and by now several hundred vibrational levels in S0, ranging beyond the first dissociation limits are known, still one of the fundamental frequencies, m6, has not yet been directly observed in the gas phase and presents a puzzle. In their work Gruebele et al. [4,11], based on a simple effective Hamiltonian (without direct resonances), successfully describing a large number of S0 vibrational states (including extremely highly excited ones), have set this frequency at m6 = 323.77 cm1. However we have serious reasons to believe, that this value is too high, based on earlier spectroscopic measurements [15,36], where m6 was estimated to lie in the range 300–305 cm1. In particular, the well known strong Fermi resonance m5 m2 + m6, observed at 792 and 818 cm1 [15], could hardly be reconciled with such a large value for m6. In addition, an earlier Raman and IR study of thiophosgene in the condensed phase by Frenzel, Blick, Bennett, and Niedenzu [16] showed clearly defined bands centered at 305 cm1. Likewise, the recent ab initio quartic fields calculated by Davisson et al. [22], at various levels of precision, have consistently yielded a m6 300 cm1. Therefore in this work for our adjustment procedure, we have decided to use the value m6 = 305 cm1, for which we hope not to be very far from reality. The other improvement to our input data comes from the analyses by McKellar and Billinghurst [17] of the m2 and m4 bands recorded at ultra high resolution with a cyclotron light source. Their rotational analyses place the origins of these two bands at m2 = 503.80679 and m4 = 471.04267 cm1. In addition they observed a weak band at 469.64 cm1 that was ascribed to a 2m4–m4 difference combination that leads to 2m4 = 940.68 cm1. More recently, McKellar [18] has assigned a weak structure in the 316 cm1 region to the ‘‘hot-band’’ transition m5–m2 by comparing the rotational constants of the lower vibrational state with those of m2 upper vibrational level. When combined with m2, this yields a value of m5 = 819.614 cm1, in agreement with the earlier direct IR observations. In Table 2 we present the results from our vibrational calculations on the 35Cl2CS isotopomer of thiophosgene for the vibrational frequencies (J = 0) of various symmetries up to 3000 cm1 that were obtained with the final adjusted set of force constants, listed
Table 2 Vibrational frequencies (in cm1) for S0 35Cl2CS with symmetries and assignments, calculated using the refined PES in this work; the experimentally measured frequencies are also displayed (column 3); the 19 frequencies that were employed in the fitting process are indicated with an asterisk (column 3); most experimentally measured data are taken from Ref. [11] and are not indicated in the Table; the remaining experimental frequencies were taken from (a) Ref. [16], (b) Ref. [17], (c) Ref. [15], (d) Ref. [18]. Assignment (symmetry)
Calculated (this work)
Experimentally observed
m3 (A1) m6 (B2) m4 (B1) m2 (A1) 2m3 (A1) 2m6 (A1) m3 + m6 (B2) m3 + m4 (B1) m4 + m6 (A2) m2 + m3 (A1) m2 + m6 (B2) m5 (B2) 3m6 (B2) m3 + 2m6(A1) 2m4 (A1) m2 + m4 (B1) 2m2(A1)
293.61 305.20 470.42 503.38 592.50 637.89 641.0 763.19 775.12 789.09 796.79 814.97 910.94 927.54 941.18 961.88 1013.72 1031.07 1034.45 1060.55 1090.68 1099.48 1103.08 1105.75 1104.97 1109.24 1117.36 1138.58 1224.75 1236.93 1246.84 1250.70 1256.06 1283.45 1299.09 1308.54 1309.23 1327.66 1332.40 1379.55 1386.69 1389.79 1394.39 1400.92 1404.93 1415.73 1418.73 1422.66 1438.27 1445.71 1455.48 1457.06 1474.70 1490.55 1494.54 1502.52 1525.51 1577.84 1639.58 1643.40 1872.83 1939.82 2073.55 2175.30 2254.23 2270.13 2351.67 2364.94
292.80⁄ 305⁄(a) 471.04267⁄(b) 503.80679⁄(b)
(B2)? (A1)? 2m3 + m4 (B1) m2 + 2m3 (A1) m2 + m3 + m6 (B2) 2m6 + m4 (B1) m3 + m4 + m6 (A2) m2 + 2m6 (A1) m3 + m5 (B2) m5 + m6 (A1) m1 (A1) 4m6 (A1) 2m4 + m3 (A1) 2m4 + m6 (B2) m2 + m3 + m4 (B1) m2 + m4 + m6 (A2) m4 + m5 (A2) 2m2 + m6 (B2) 2m2 + m3 (A1) m2 + m5 (B2) 2m2 + m6 (B2) 2m2 + m3(A1) m4 + 3m6 (A2) 2m3 + m5 (B2) 3m4 (B1) m2 + m3 + 2m6(A1) m3 + m5 + m6 (A1) m5 + 2m3 (B2) m1 + m3 (A1) m2 + 2m4 (A1) m5 + 2m6 (B2) m1 + m6 (B2) 5m3 (A1) 3m3 + 2m6 (A1) 4m3 + m6 (B2) 2m2 + m4 (B1) 2m3 + 3m6 (B2) 3m2 (A1) (A2)? 2m3 + 2m4 (A1) m3 + m4 + m5 (A2) 2m5 (A1) m1 + m2 (A1) 4m4 (A1) 2m2 + 2m4 (A1) m1 + 2m4 (A1) m3 + 4m4 (A1) m1 + m5 + m6 (A1) 2m1 (A1) m3 + 2m4 + m5 + m6 (A1) m1 + m3 + 2m4 (A1)
792.0(c) 819.614⁄(d)
940.68⁄(b)
1139.70⁄
1425.0(c)
1442.80⁄
1644.67⁄ 1881.00⁄ 1940.17⁄ 2071.87⁄ 2176.00⁄ 2256.33 2267.93⁄ 2351.97 2364.07⁄
S. Rashev, D.C. Moule / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 140 (2015) 305–310 Table 2 (continued) Assignment (symmetry)
Calculated (this work)
Experimentally observed
m2 + 4m4 (A1) 2m3 + 4m4 (A1) m1 + m2 + 2m4 (A1) m1 + 2m2 + 2m6 (A1) m1 + 2m5 (A1) 2m1 + m2 (A1) 6m4 (A1) m1 + m5 + 3m6 (A1) 2m2 + 4m4 (A1) m1 + 4m4 (A1) m1 + 2m2 + 2m4 (A1) m3 + 6m4 (A1) 2m1 + m3 + 2m6 (A1)
2378.94 2468.54 2571.41 2751.23 2758.15 2771.48 2822.57 2863.65 2872.88 3003.13 3065.25 3115.08 3178.25
2380.27 2468.80⁄ 2572.43⁄ 2752.80 2758.00⁄ 2771.37⁄ 2822.33⁄ 2866.50 2876.10 3005.53 3068.27 3114.53 3180.13
Table 3 A set of adjusted quartic force constants for S0 35Cl2CS, determined in this work (units are aJ and Radians; stretches are in dimensionless SPF coordinates), using our vibrational code [equilibrium configuration: r(C–S) = 1.607 Å, r(C–Cl) = 1.733 Å, \(Cl– C–Cl) = 111.32°]. F11 = 6.2479 F22 = 8.2978 F33 = 1.4168 F44 = 0.13858 F55 = 3.7321 F66 = 0.48597 F12 = 2.4318 F13 = 1.5615 F23 = 0.44083 F56 = 0.4636 F111 = 5.6705 F112 = 13.6105 F113 = 1.5999 F122 = 0.66951 F123 = 3.6344 F133 = 4.2436 F144 = 0.95929 F155 = 9.2635 F156 = 1.4354 F166 = 1.4741 F222 = 8.3037 F223 = 1.6590 F233 = 0.51692 F244 = 0.31557 F255 = 6.9852 F256 = 3.8819 F266 = 0.89195 F333 = 1.2341 F344 = 0.17235 F355 = 4.0610 F356 = 1.0886 F366 = 1.0196 F1111 = 1.3109 F1112 = 0.0311 F1113 = 1.3784 F1122 = 0.64930 F1123 = 3.0429 F1133 = 4.7055 F1144 = 0.0100 F1155 = 4.4703 F1156 = 3.1764 F1166 = 0.034596
F1222 = 4.4995 F1223 = 1.8702 F1233 = 1.2449 F1244 = 5.9400 F1255 = 2.2492 F1256 = 0.16168 F1266 = 1.7459 F1333 = 5.2725 F1344 = 1.1745 F1355 = 4.7744 F1356 = 0.67549 F1366 = 1.0701 F2222 = 4.8746 F2223 = 5.2352 F2233 = 0.10819 F2244 = 1.1074 F2255 = 1.9943 F2256 = 6.2071 F2266 = 0.013498 F2333 = 0.32811 F2344 = 0.33798 F2355 = 3.2184 F2356 = 3.5266 F2366 = 1.3568 F3333 = 2.5758 F3344 = 0.092335 F3355 = 5.3227 F3356 = 2.3066 F3366 = 0.72612 F4444 = 0.046084 F4455 = 5.4569 F4456 = 0.79065 F4466 = 0.19130 F5555 = 0.012798 F5556 = 0.11605 F5566 = 0.23490 F5666 = 0.45897 F6666 = 0.10688 F33344 = 5.563721 F34466 = 3.1854347
in Table 3. The experimentally measured frequencies and assignments [11] are also displayed in Table 2. The vibrational frequencies used in the adjustment procedure, have been denoted by an asterisk. All calculated vibrational frequencies up to 1500 cm1, with our assignments, have been listed in Table 2. Next, in the range 1500–3000 cm1, only those calculated levels, that correspond to the experimentally measured frequencies and their
309
assignments have been displayed. We have provided our own tentative assignments for those levels in the whole energy range, that have been observed in Ref. [11], but not assigned by the authors. From a comparison of the calculated with the spectroscopically measured frequencies it can be concluded, that our refined PES is a reasonable representation to the molecular S0 potential field. In particular, the m5 m2 + m6, Fermi resonance components have been satisfactorily reproduced. The remaining most prominent resonance pairs m1 m5 + m6, m1 + m2 2m5, 2m3 2m6 have also been obtained at reasonable locations (1138, 1117 cm1), (1643, 1639 cm1), and (592, 637 cm1), respectively. Of interest are the values for m6 and 2m6 of 305.20 and 637.89 cm1 contained in Table 2. As two times the m6 fundamental frequency is 610.40 cm1, the m6 manifold is highly anharmonic. The increasing value of the vibrational intervals with increasing energy can be attributed to a flattening of the potential resulting from the contribution of the positive F6666 and F3366 quartic oscillator terms. These observations lead to our suspicion that Gruebele’s [4] assignment of the intervals of 646.8 cm1 in the resolved fluorescence spectrum to overtones of m6 may indeed be correct and that vibrational anharmonicity may be responsible for the discrepancies between the various values for m6. In Table 3 the adjusted values of all quadratic, cubic and quartic force constants included in the PES of thiophosgene (Eq. (1)) are presented, as well as the optimized values of the three equilibrium parameters (two bond lengths and one interbond angle). In the Supplement to this work we have provided a C++ code for calculation of the potential energy for arbitrary input values of the displacements of all six vibrational coordinates from equilibrium. It will be our next aim, using the obtained quartic PES for thiophosgene, to perform calculations in the range of much higher excited vibrational levels in order to explore the vibrational mixing and IVR behavior, based on a realistic 6D potential field. Acknowledgement This research was supported by the National Research and Engineering Council of Canada. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.saa.2014.12.101. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
A. Maciejewski, R.P. Steer, Chem. Rev. 93 (1993) 67. D.C. Moule, I.R. Burling, H. Liu, E.C. Lim, J. Chem. Phys. 111 (1999) 5027. R. Bigwood, B. Milam, M. Gruebele, Chem. Phys. Lett. 287 (1998) 333. B. Strickler, M. Gruebele, Phys. Chem. Chem. Phys. 6 (2004) 3786. D.C. Moule, E.C. Lim, J. Chem. Phys. 110 (1999) 9341. T. Fujiwara, D.C. Moule, E.C. Lim, J. Phys. Chem. A 107 (2003) 10223. T. Fujiwara, E.C. Lim, R.H. Judge, D.C. Moule, J. Chem. Phys. 124 (2006) 124301. S. Lee, M. Engel, M. Gruebele, Chem. Phys. Lett. 420 (2006) 151. E.L. Sibert, M. Gruebele, J. Chem. Phys. 124 (2006) 024317. C. Jung, H.S. Taylor, E.L. Sibert, J. Phys. Chem. A 110 (2006) 5317. E. Berrios, S. Pratt, P. Tripathi, M. Gruebele, J. Phys. Chem. A 117 (2013) 12082– 12090. S. Keshavamurthy, J. Chem. Sci. 124 (2012) 291–300. R.J. Lowell, E.A. Jones, J. Mol. Spectrosc. 4 (1960) 173–189. J. Downs, Spectrochim. Acta 19 (1963) 1165–1171. J.C.D. Brand, J.H. Callomon, D.C. Moule, J. Tyrrell, T.H. Goodwin, Trans. Faraday Soc. 61 (1965) 2365. C.A. Frenzel, K.E. Blick, C.R. Bennett, K. Niedenzu, J. Chem. Phys. 53 (1970) 198– 204. A.R.W. McKellar, B.E. Billinghurst, J. Mol. Spectrosc. 260 (2010) 66–71. A.R.W. McKellar, private communication. P.D. Chowdary, B. Strickler, S. Lee, M. Gruebele, Chem. Phys. Lett. 434 (2007) 182–187. P.D. Chowdary, M. Gruebele, J. Chem. Phys. 130 (2009) 024305. S. Rashev, I. Bivas, D.C. Moule, Chem. Phys. Lett. 438 (2007) 153–156. J.L. Davisson, N.R. Brinkmann, W.F. Polik, Mol. Phys. 110 (2012) 2587–2598.
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[23] J.M.L. Martin, T.J. Lee, P.R. Taylor, J. Mol. Spectrosc. 160 (1993) 105. [24] N.C. Handy, Mol. Phys. 61 (1987) 207. [25] S. Rashev, D.C. Moule, Spectrochim. Acta A, Mol. Biomol. Spectrosc. 87 (2012) 286–292. [26] S. Rashev, D.C. Moule, V. Rashev, Spectrochim. Acta A, Mol. Biomol. Spectrosc. 97 (2012) 111–118. [27] S. Rashev, D.C. Moule, J. Mol. Spectrosc. 286–287 (2013) 21–29. [28] S. Rashev, D.C. Moule, in: Advances in Quantum Methods and Applications in Chemistry, Physics, and Biology, Prog. Theoret. Chem. Phys. B, vol. 27, Springer, 2013, pp. 141–160 (Chapter 8). [29] G. Simons, R.G. Parr, J.M. Finlan, J. Chem. Phys. 59 (1973) 3229.
[30] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C, Cambridge University Press, Cambridge, 1988. [31] J.K. Cullum, R.A. Willowghby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, vols. I and II, Birkhauser, Boston, 1985. [32] R.E. Wyatt, Adv. Chem. Phys. 73 (1989) 231. [33] G.M. Stewart, J.D. McDonald, J. Chem. Phys. 78 (1983) 3907. [34] M. Valiev, E.J. Bylaska, N. Govind, K. Kowalski, T.P. Straatsma, H.J.J. van Dam, D. Wang, J. Nieplocha, E. Apra, T.L. Windus, W.A. de Jong, Comput. Phys. Commun. 181 (2010) 1477. [35] A.R. Brazzale, Rnews, 5/1 May 2005. [36] D.J. Clouthier, D.C. Moule, Top. Curr. Chem. 150 (1989) 167.
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Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa
A refined quartic potential energy surface and large scale vibrational calculations for S0 thiophosgene Svetoslav Rashev a,⇑, David C. Moule b a b
Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradskochaussee 72, 1784 Sofia, Bulgaria Department of Chemistry, Brock University, St. Catharines, ON L2S3A1, Canada
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
We compute an ab initio quartic force
field for S0 thiophosgene Cl2CS. We calculate vibrational energy levels
using our specific variational method. We carry out an adjustment
procedure on the ab initio computed quartic PES. We calculated a large number of excited vibrational level energies. We obtained a good fit between calculated and experimental frequencies.
a r t i c l e
i n f o
Article history: Received 13 October 2014 Received in revised form 8 December 2014 Accepted 28 December 2014 Available online 6 January 2015 Keywords: Thiophosgene Potential energy surface Refined surface Variational vibrational calculations Experimental frequencies
a b s t r a c t In this work we present a full 6D quartic potential energy surface (PES) for S0 thiophosgene in curvilinear symmetrized bond-angle coordinates. The PES was refined starting from an ab initio field derived from acc-pVTZ basis set with CCSD(T) corrections for electron correlation. In the present calculations we used our variational method that was recently tested on formaldehyde and some of its isotopomers, along with additional improvements. The lower experimentally known vibrational levels for 35Cl2CS were reproduced quite well in the calculations, which can be regarded as a test for the feasibility of the obtained quartic PES. Ó 2015 Elsevier B.V. All rights reserved.
Introduction Thiophosgene is an interesting molecule, that has attracted considerable attention both experimentally as well as theoretically, as it has proved to be a very suitable model system for studying of a large variety of generic spectral and photophysical manifestations in polyatomic molecules [1–12]. Thiophosgene has been recognized both as a suitable molecule for studying ‘‘backbone’’ intramolecular vibrational energy redistribution (IVR) in the ground ⇑ Corresponding author. E-mail address: [email protected] (S. Rashev). http://dx.doi.org/10.1016/j.saa.2014.12.101 1386-1425/Ó 2015 Elsevier B.V. All rights reserved.
electronic state S0 [3,4,8,9], as well as ‘‘a molecule, tailor-made for studying fundamental concepts of electronic radiationless transitions’’ [1,2,5–7]. The vibrational spectroscopy of thiophosgene has also been studied in considerable detail, although even at the present time some vibrational fundamentals have not been directly observed. As a result of an extensive research effort, many ground electronic state frequencies have been measured up to extremely high vibrational excitation energies, of 23,000 cm1, using low resolution IR [13–16] and high resolution synchrotron IR spectroscopy [17], laser induced fluorescence and stimulated emission pumping from both excited electronic states S1 and S2 [4,8,11,19,20]. This range of observed vibrational frequencies
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extends up to and even exceeds the first two dissociation limits (19,900 cm1 to CS + Cl2, and 22,200 cm1 to Cl + SCCl). A surprisingly large number of the spectroscopically observed highly excited vibrational levels (several hundred), including many levels above dissociation limit, have been successfully observed and assigned using a simple nonresonant effective Hamiltonian expression by Gruebele and collaborators with only 20 adjustable parameters [4,11]. This suggests, that characteristic vibrational feature states – well isolated quantum states with assignable quantum number composition – should exist in the whole range of S0 vibrational excitation and even above the lowest dissociation limits [11]. Many theoretical models, both quantum mechanical and semi-classical, have been designed and applied to the understanding of the extent and mechanisms of vibrational level mixing and vibrational energy redistribution (IVR) in S0 thiophosgene [4,9– 12]. In particular, rationalizing the existence of numerous highly excited assignable vibrational states, not strongly affected by vibrational mixing – which contradicts simple statistical theories – is a major theoretical challenge. The existence of such states suggests the possibility for controlled monomolecular reactions, through the selective initial excitation of appropriate vibrational levels. Several model 6D potential fields for S0 thiophosgene have been calculated so far in the literature [4,21,22]. The ab initio field of Strickler and Gruebele [4] has been calculated with bond distance coordinates and then refined using over 100 vibrational levels up to 8000 cm1. The recent quartic field by Davisson et al. [22] was computed using a CCST(T) method, in terms of symmetrized internal coordinates, and then transformed into Cartesian coordinates. The authors’ objective in this work was to show, that it was highly advantageous computationally to use hybrid basis sets in the quantum mechanical computations, namely an aug-cc-pVQZ for the equilibrium geometry and harmonic force field, while a ccpVTZ for the cubic and quartic force constants. The hybrid approach albeit highly economic, was shown to be of only slightly inferior quality as compared to the use of an aug-cc-pVQZ throughout. For their vibrational calculations, Davisson et al. [22] used the so called VPT2+K approach, which is based on second order vibrational perturbation theory with resonances treated explicitly using a Hamiltonian formalism. Our earlier empirical field [21], was of limited scope and applicability since our vibrational calculation method involved a simplified (approximate) expression for the kinetic energy operator and in addition the field included only the quadratic and several (small part) of the cubic and quartic force constants. In this work we present a full quartic PES for S0 thiophosgene (including all 34 quadratic, cubic and quartic force constants), in terms of the symmetrized curvilinear coordinates that were used by MLT [23] and Handy [24] in their work on formaldehyde (different from the coordinates used in our previous work [21] and from those used by Strickler and Gruebele [4]). The field obtained in this work appears to be of good quality, as it was able to reproduce the experimentally measured lower excited vibrational levels (fundamentals, overtones and combinations) in S0 thiophosgene. Using this field and our recently developed variational method, we carry out detailed large scale calculations on the vibrational level structure of S0 thiophosgene up to 3000 cm1. We calculate vibrational levels of all symmetries and compare our results to the experimentally observed frequencies in this energy range. In most cases we provide our own assignments and compare them to the assignments by other authors. We also discuss the most important resonance interactions (of Fermi and Darling-Dennison type). In forthcoming work, using the obtained PES, it will be our purpose to explore the vibrational mixing patterns in S0 thiophosgene at very high vibrational excitation and perhaps check the validity of the restricted vibrational models, that have been employed by other authors for this purpose.
Vibrational Hamiltonian and vibrational basis set We employ our Hamiltonian formalism, Hvib = T + V, that we recently developed and described in detail for the exploration of the vibrational level structure and vibrational level mixing in S0 formaldehyde and isotopomers [25–28]. The kinetic energy expression T is the same exact expression (with changed nuclear masses) as used for formaldehyde (Handy [24]), while the quartic PES V is described as a quartic Taylor expansion [23],
V ¼ E0 þ
X X X F ij Q i Q j þ F ijk Q i Q j Q k þ F ijkl Q i Q j Q k Q l ; ij
ijk
ð1Þ
ijkl
In terms of 6 symmetrized curvilinear coordinates Qi, i = 1, . . ., 6, defined as follows (C2v symmetry species are indicated): p p Q1(A1) = qC–S, Q2(A1) = (q1 + q2)/ 2, Q3(A1) = (h1 + h2)/ 2, Q4(B1) = p p sCl–C–S–Cl0 , Q5(B2) = (q1 q2)/ 2, Q6(B2) = (h1 h2)/ 2. Here q1,2 = Dr1,2/(Dr1,2 + r0), qC–S = DrC–S/(DrC-S + r0,C–S) are three Simmons–Parr–Finlan (SPF) coordinates [29] for the two stretches Dr1,2 from equilibrium r0 of the two Cl–C bonds, and the stretch DrC–S from equilibrium r0,C–S of the C–S bond, respectively; h1 and h2 are the two bends from equilibrium h0,Cl–C–S of the two Cl–C–S angles and sCl–C–S–Cl0 is the change of the ‘‘book’’ angle between planes Cl1–C–S and Cl2–C–S, from equilibrium p. Both kinetic energy [24] and potential energy (1) expressions are presented in separable form (sum of products of 1D functions), therefore the 6D basis wavefunctions that form the primitive basis set, can conveniently be used in the form of products of 1D basis functions corresponding to each of the local vibrational coordinates: q1,2, qC–S, h1,2, sCl–C–S–Cl0 . In the following we shall briefly specify our definition and construction of the 1D basis sets for each of the curvilinear coordinates. The basis functions for each of the three stretches Cl1–C, Cl2–C and C–S respectively, are originally taken as 31 Morse oscillator eigenfunctions, |n1i, |n2i, |n3i, n1, n2, n3 = 0, . . ., 30. With a suitable choice of the Morse parameters (harmonic frequency and anharmonicity corresponding to the type of molecular vibration described), these sets of basis functions can be adapted to give a distribution of energy levels En = hn|Hvib|ni that roughly corresponds to the relevant molecular fundamental and overtone vibrational frequencies. This is the type of basis that we have employed in our recent calculations on formaldehyde and isotopomers [25–28]. However these basis functions have large and increasing with nnondiagonal matrix elements hn|Hvib|n+1i, which exceed the energy separation of adjacent levels En+1 En, at the larger n values. This is not convenient for the operation of our search selection algorithm, especially when exploring very highly excited overtone excitations. Therefore we performed optimization on the Morse functions, by prediagonalization with a simple appropriate 1D Hamiltonian, which yields a new set of basis wavefunctions, that are free from the above mentioned defect. These basis functions will be denoted from now on as |n1i, |n2i, |n3i, n1, n2, n3 = 0, . . ., 30, for each the stretches Cl1–C, Cl2–C and C–S, respectively (31 basis functions of each type). For the out of plane bend, we employ 38 harmonic oscillator eigenfunctions |n4i, without additional prediagonalization. Finally, for the two S–C–Cl(h) bends we start with a set of normalized associated Legendre polynomials P 2n ðcos hÞ, n = 2, 3, . . . that cancel the singularities in the KE operator T. These wavefunctions have no free parameters to adjust and are not well adapted to the molecular vibrational levels, therefore for them we also apply a prediagonalization of the 1D basis using an appropriate simple Hamiltonian, as described in detail recently [26]. The basis functions that are obtained from this procedure, are denoted as |n5i, |n6i (n5,n6 = 0, . . ., 48), for the two bending modes h1,2, respectively. As a result of the separable form of the Hamiltonian, a 6D basis function can be presented in product form: |n1, n2, n3, n4, n5, n6i = |n1i |n2i |n3i |n4i |n5i |n6i and a Hamiltonian
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matrix element hn1, n2, n3, n4, n5, n6|Hvib|n01 , n02 , n03 , n04 , n05 , n06 i is reduced to a sum of products of factors, that are obtained through 1D numerical integration. The dimensionality of the primitive basis Q set is N p ¼ 6i¼1 n0i ¼ 1; 608; 161; 632. To accelerate the large scale calculations, prior to each actual vibrational calculation, we i;a compute a number of 2D arrays Pni ;ni 0 ¼ hni ðqi ÞjF ai ðqi Þjn0i ðqi Þi, i 0 ni, ni = 0, 1, 2, . . ., n0,i, for each vibrational coordinate qi and each function or operator depending on this coordinate F ai ðqi Þ, occurring in either KE or PES expressions, using either Gauss–Hermite, Gauss–Laguerre or Gauss–Legendre numerical integration [30], where n0,i is the number of basis functions employed for each vibrational coordinate qi. All computed n0,i n0,i0 arrays are stored in computer core memory, ready to use in the subsequent matrix elements calculations. Now, each matrix element can be obtained ai as the sum of products of the appropriate P i;m;n values hence reducing the actual calculation to a series of multiplications and summations and no integrations, which greatly accelerates the calculation of matrix elements. In our calculations we employ symmetrized basis functions |n1, n2, n3, n4, n5, n6; Si, that are obtained as linear combinations of the original product wavefunctions |n1, n2, n3, n4, n5, n6i. E.g., if in a product state one or both pairs of degenerate modes have different quantum numbers (n1 – n2 and/or n5 – n6), we symmetrize by taking |n1, n2, n3, n4, n5, n6; Si = (|n1, n2, n3, n4, n5, n6i ± |n2, n1, n3, n4, n6, n5p 3, n4, n5, n6; Si = (|n1, n2, n3, n4, n5, n6i ± |n2, n1, n3, n4, n6, n5i)/ 2, where symmetry can be S = A1,B2, if n4 – even or S = B1,A2, if n4 – uneven. Thus, our basis functions are not the usual normal modes, but they are superpositions of symmetrized local modes excitations.
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the selected AS (for increasing R and decreasing C values), which will result in enhanced accuracy and display the convergence in the calculation of the desired molecular vibrational levels. All selected basis states are stored as an array in computer core memory. Simultaneously the Hamiltonian matrix is built that contains the diagonal and nondiagonal Hamiltonian matrix elements of the selected basis states. The Hamiltonian matrix H constructed in the course of the search/selection procedure, besides being optimal in size, is also quite sparse, because the algorithm employed automatically discards the matrix elements that are too small according to the criteria of the search. This makes our vibrational variational procedure both memory and time efficient. For the tridiagonalization of H we employ a conventional Lanczos iteration without reorthogonalization [31,32], starting with the vector |0i. We diagonalize the resulting tridiagonal Lanczos matrix using the routine tqli() from numerical recipes [30], in slightly modified form. From this calculation we obtain the spectral distribution |Ci|2 in the explored range of vibrational excitations, where Ci is the first component (corresponding to basis state |0i) of the eigenvector |ii, at eigenenergy Ei. The dilution factor r [33] is a measure of the vibrational fragP mentation around |0i and is given by the expression: r ¼ k jC k j4 [9]. In fact, Neff = r1 accounts for the average number of vibrational eigenstates |ii that are effectively intermixed with (coupled to) the feature (or ‘‘bright’’) state |0i i.e., it may be considered to represent the extent of vibrational mixing around |0i.
Variational vibrational calculation procedure for thiophosgene
Calculation of an ab initio quartic PES for thiophosgene and adjustment of the ab initio calculated PES to the experimentally measured frequencies
Our specific search/selection procedure [25] serves to select an optimally small however representative active space (AS) of NAS basis states (from a huge available primitive space Np), that are most relevant to the particular vibrational calculation and to simultaneously construct the relevant Hamiltonian matrix H. A search/selection procedure is started from a particular basis (feature) state |0i, chosen to be the best zeroth-order representation of the vibrational levels that we are interested in. The algorithm is symmetrically adapted to search and select only such basis states whose symmetry coincides with the symmetry species of the initial state |0i. During the implementation of the search/selection algorithm, many basis states are being probed and each state that satisfies the criteria for sufficient coupling strength is selected and consecutively added to the previously selected AS. Although the employed optimized search/selection algorithm is in itself a rather complicated sequence of procedures, there are essentially two major parameters, determining the scope and the quality of the search, C and R (whose values have to be fixed at the outset), that have been defined and discussed in our previous work [25]. R (a filtering parameter, usually chosen as 1000 cm1) is a window around the zeroth energy of the initial (for the search) basis state |0i, which acts to promote the selection of basis states that are located within the energetic R vicinity of |0i and to prevent the selection of too strongly displaced basis states. An evaluation function value fk is ascribed to each probed for selection basis state |ki, accounting for its (both direct and indirect) coupling strength to |0i. According to its definition [25], fk is 1 for the initial state |0i and attains smaller and diminishing (but positive) values for all subsequently selected states (R is also involved in the determination of fk). C (usually chosen as 1010 or smaller) is the limiting value for fk. Probed basis states with fk < C will not be selected. According to the values chosen for the parameters C and R, the search/selection procedure will select a varying number of basis states, i.e., include more and more weakly coupled basis states into
First, a full quartic potential field for thiophosgene was generated with the nwchem suite of ab initio programs [34]. The approximation used for this PES study was the cc-p VTZ correlation consistent polarized triple valence basis. This was further corrected for electron correlation by CCSD(T). This is a coupled cluster method with single and double substitutions augmented by a perturbative estimate of the triple excitations. To create the grid of structural points as input for the energy calculations, the bond lengths were adjusted in step sizes of 0.05, 0.1, 0.2 Å, while the bond angles were varied by increments of 5° and 10°. The potential energy function was fitted to 1083 resulting data points with the nonlinear regression package NLREG [35] with a mean square error of 36 cm1. In Table 1 we present the results for the 6 fundamental 35 Cl2CS frequencies, calculated from the set of ab initio computed force constants in this work (column 3) and compared to the experimental data (column 2) and those calculated by Davisson et al. [22], using their ab initio computed hybrid basis set (column 4). From a comparison of the vibrational data, obtained using both ab initio fields with the experimental values it can be seen, that our ab initio quartic force field is somewhat inferior to that of Davisson et al. [22]. Next, the ab initio calculated quadratic, cubic and quartic force constants were refined, using our large scale variational method and a nonlinear least squares fitting algorithm (Marquardt’s method for nonlinear parameter estimation through the ‘‘chisquare’’ minimization, implemented in the routine mrqmin() from [30]. Our refinement procedure was similar to that described in our recent work for formaldehyde [28]. In addition to the force constants, the three equilibrium parameters (two bond lengths and one interbond angle) were also varied. We compared the calculated vibrational frequencies, to a selected set of 19 experimentally measured values in the lower excitation range, up to 3000 cm1. We carried out careful analysis of the convergence by varying the parameters of the search (decreasing the value of
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Table 1 Fundamental 35Cl2CS frequencies, calculated from the set of ab initio computed force constants in this work (column 3) and compared to the experimental data (column 2) and those calculated by Davisson et al. [22], using their ab initio computed hybrid basis set (column 4); our frequencies calculated from the refined PES, are given in column 5; the experimental frequencies (column 2) were taken from (a) Ref. [11], (b) Ref. [16], (c) Ref. [17], (d) Ref. [18]. Fundamental frequency
m1 (A1) m2 (A1) m3 (A1) m4 (B1) m5 (B2) m6 (B2)
Experiment
1139.70(a) 503.80679(c) 292.80(a) 471.04267(c) 819.614(d) 305(b)
Calculated ab initio (this work)
Calculated ab initio in [22] (hybrid basis set)
Calculated from the refined PES
1134.92 500.03 303.50 469.85 812.87 309.11
1138.5 502.7 292.5 469.8 820.6 301.8
1138.58 503.38 293.61 470.42 814.97 305.20
C and increasing the value of R led to enhanced accuracy of the calculated frequencies, but to increased calculation times, as well). It was found, that a calculation with C = 1012 and R = 1500 cm1 is sufficient to achieve an accuracy of 0.05 cm1 for the lower excited thiophosgene frequencies in the range 0–3000 cm1. This resulted in selected active spaces of about NAS = 30,000–50,000 basis states. In our least squares fitting procedure we varied the quadratic force constants first, before turning to the higher order constants. Although thiophosgene has been the object of numerous spectroscopic studies by many authors and by now several hundred vibrational levels in S0, ranging beyond the first dissociation limits are known, still one of the fundamental frequencies, m6, has not yet been directly observed in the gas phase and presents a puzzle. In their work Gruebele et al. [4,11], based on a simple effective Hamiltonian (without direct resonances), successfully describing a large number of S0 vibrational states (including extremely highly excited ones), have set this frequency at m6 = 323.77 cm1. However we have serious reasons to believe, that this value is too high, based on earlier spectroscopic measurements [15,36], where m6 was estimated to lie in the range 300–305 cm1. In particular, the well known strong Fermi resonance m5 m2 + m6, observed at 792 and 818 cm1 [15], could hardly be reconciled with such a large value for m6. In addition, an earlier Raman and IR study of thiophosgene in the condensed phase by Frenzel, Blick, Bennett, and Niedenzu [16] showed clearly defined bands centered at 305 cm1. Likewise, the recent ab initio quartic fields calculated by Davisson et al. [22], at various levels of precision, have consistently yielded a m6 300 cm1. Therefore in this work for our adjustment procedure, we have decided to use the value m6 = 305 cm1, for which we hope not to be very far from reality. The other improvement to our input data comes from the analyses by McKellar and Billinghurst [17] of the m2 and m4 bands recorded at ultra high resolution with a cyclotron light source. Their rotational analyses place the origins of these two bands at m2 = 503.80679 and m4 = 471.04267 cm1. In addition they observed a weak band at 469.64 cm1 that was ascribed to a 2m4–m4 difference combination that leads to 2m4 = 940.68 cm1. More recently, McKellar [18] has assigned a weak structure in the 316 cm1 region to the ‘‘hot-band’’ transition m5–m2 by comparing the rotational constants of the lower vibrational state with those of m2 upper vibrational level. When combined with m2, this yields a value of m5 = 819.614 cm1, in agreement with the earlier direct IR observations. In Table 2 we present the results from our vibrational calculations on the 35Cl2CS isotopomer of thiophosgene for the vibrational frequencies (J = 0) of various symmetries up to 3000 cm1 that were obtained with the final adjusted set of force constants, listed
Table 2 Vibrational frequencies (in cm1) for S0 35Cl2CS with symmetries and assignments, calculated using the refined PES in this work; the experimentally measured frequencies are also displayed (column 3); the 19 frequencies that were employed in the fitting process are indicated with an asterisk (column 3); most experimentally measured data are taken from Ref. [11] and are not indicated in the Table; the remaining experimental frequencies were taken from (a) Ref. [16], (b) Ref. [17], (c) Ref. [15], (d) Ref. [18]. Assignment (symmetry)
Calculated (this work)
Experimentally observed
m3 (A1) m6 (B2) m4 (B1) m2 (A1) 2m3 (A1) 2m6 (A1) m3 + m6 (B2) m3 + m4 (B1) m4 + m6 (A2) m2 + m3 (A1) m2 + m6 (B2) m5 (B2) 3m6 (B2) m3 + 2m6(A1) 2m4 (A1) m2 + m4 (B1) 2m2(A1)
293.61 305.20 470.42 503.38 592.50 637.89 641.0 763.19 775.12 789.09 796.79 814.97 910.94 927.54 941.18 961.88 1013.72 1031.07 1034.45 1060.55 1090.68 1099.48 1103.08 1105.75 1104.97 1109.24 1117.36 1138.58 1224.75 1236.93 1246.84 1250.70 1256.06 1283.45 1299.09 1308.54 1309.23 1327.66 1332.40 1379.55 1386.69 1389.79 1394.39 1400.92 1404.93 1415.73 1418.73 1422.66 1438.27 1445.71 1455.48 1457.06 1474.70 1490.55 1494.54 1502.52 1525.51 1577.84 1639.58 1643.40 1872.83 1939.82 2073.55 2175.30 2254.23 2270.13 2351.67 2364.94
292.80⁄ 305⁄(a) 471.04267⁄(b) 503.80679⁄(b)
(B2)? (A1)? 2m3 + m4 (B1) m2 + 2m3 (A1) m2 + m3 + m6 (B2) 2m6 + m4 (B1) m3 + m4 + m6 (A2) m2 + 2m6 (A1) m3 + m5 (B2) m5 + m6 (A1) m1 (A1) 4m6 (A1) 2m4 + m3 (A1) 2m4 + m6 (B2) m2 + m3 + m4 (B1) m2 + m4 + m6 (A2) m4 + m5 (A2) 2m2 + m6 (B2) 2m2 + m3 (A1) m2 + m5 (B2) 2m2 + m6 (B2) 2m2 + m3(A1) m4 + 3m6 (A2) 2m3 + m5 (B2) 3m4 (B1) m2 + m3 + 2m6(A1) m3 + m5 + m6 (A1) m5 + 2m3 (B2) m1 + m3 (A1) m2 + 2m4 (A1) m5 + 2m6 (B2) m1 + m6 (B2) 5m3 (A1) 3m3 + 2m6 (A1) 4m3 + m6 (B2) 2m2 + m4 (B1) 2m3 + 3m6 (B2) 3m2 (A1) (A2)? 2m3 + 2m4 (A1) m3 + m4 + m5 (A2) 2m5 (A1) m1 + m2 (A1) 4m4 (A1) 2m2 + 2m4 (A1) m1 + 2m4 (A1) m3 + 4m4 (A1) m1 + m5 + m6 (A1) 2m1 (A1) m3 + 2m4 + m5 + m6 (A1) m1 + m3 + 2m4 (A1)
792.0(c) 819.614⁄(d)
940.68⁄(b)
1139.70⁄
1425.0(c)
1442.80⁄
1644.67⁄ 1881.00⁄ 1940.17⁄ 2071.87⁄ 2176.00⁄ 2256.33 2267.93⁄ 2351.97 2364.07⁄
S. Rashev, D.C. Moule / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 140 (2015) 305–310 Table 2 (continued) Assignment (symmetry)
Calculated (this work)
Experimentally observed
m2 + 4m4 (A1) 2m3 + 4m4 (A1) m1 + m2 + 2m4 (A1) m1 + 2m2 + 2m6 (A1) m1 + 2m5 (A1) 2m1 + m2 (A1) 6m4 (A1) m1 + m5 + 3m6 (A1) 2m2 + 4m4 (A1) m1 + 4m4 (A1) m1 + 2m2 + 2m4 (A1) m3 + 6m4 (A1) 2m1 + m3 + 2m6 (A1)
2378.94 2468.54 2571.41 2751.23 2758.15 2771.48 2822.57 2863.65 2872.88 3003.13 3065.25 3115.08 3178.25
2380.27 2468.80⁄ 2572.43⁄ 2752.80 2758.00⁄ 2771.37⁄ 2822.33⁄ 2866.50 2876.10 3005.53 3068.27 3114.53 3180.13
Table 3 A set of adjusted quartic force constants for S0 35Cl2CS, determined in this work (units are aJ and Radians; stretches are in dimensionless SPF coordinates), using our vibrational code [equilibrium configuration: r(C–S) = 1.607 Å, r(C–Cl) = 1.733 Å, \(Cl– C–Cl) = 111.32°]. F11 = 6.2479 F22 = 8.2978 F33 = 1.4168 F44 = 0.13858 F55 = 3.7321 F66 = 0.48597 F12 = 2.4318 F13 = 1.5615 F23 = 0.44083 F56 = 0.4636 F111 = 5.6705 F112 = 13.6105 F113 = 1.5999 F122 = 0.66951 F123 = 3.6344 F133 = 4.2436 F144 = 0.95929 F155 = 9.2635 F156 = 1.4354 F166 = 1.4741 F222 = 8.3037 F223 = 1.6590 F233 = 0.51692 F244 = 0.31557 F255 = 6.9852 F256 = 3.8819 F266 = 0.89195 F333 = 1.2341 F344 = 0.17235 F355 = 4.0610 F356 = 1.0886 F366 = 1.0196 F1111 = 1.3109 F1112 = 0.0311 F1113 = 1.3784 F1122 = 0.64930 F1123 = 3.0429 F1133 = 4.7055 F1144 = 0.0100 F1155 = 4.4703 F1156 = 3.1764 F1166 = 0.034596
F1222 = 4.4995 F1223 = 1.8702 F1233 = 1.2449 F1244 = 5.9400 F1255 = 2.2492 F1256 = 0.16168 F1266 = 1.7459 F1333 = 5.2725 F1344 = 1.1745 F1355 = 4.7744 F1356 = 0.67549 F1366 = 1.0701 F2222 = 4.8746 F2223 = 5.2352 F2233 = 0.10819 F2244 = 1.1074 F2255 = 1.9943 F2256 = 6.2071 F2266 = 0.013498 F2333 = 0.32811 F2344 = 0.33798 F2355 = 3.2184 F2356 = 3.5266 F2366 = 1.3568 F3333 = 2.5758 F3344 = 0.092335 F3355 = 5.3227 F3356 = 2.3066 F3366 = 0.72612 F4444 = 0.046084 F4455 = 5.4569 F4456 = 0.79065 F4466 = 0.19130 F5555 = 0.012798 F5556 = 0.11605 F5566 = 0.23490 F5666 = 0.45897 F6666 = 0.10688 F33344 = 5.563721 F34466 = 3.1854347
in Table 3. The experimentally measured frequencies and assignments [11] are also displayed in Table 2. The vibrational frequencies used in the adjustment procedure, have been denoted by an asterisk. All calculated vibrational frequencies up to 1500 cm1, with our assignments, have been listed in Table 2. Next, in the range 1500–3000 cm1, only those calculated levels, that correspond to the experimentally measured frequencies and their
309
assignments have been displayed. We have provided our own tentative assignments for those levels in the whole energy range, that have been observed in Ref. [11], but not assigned by the authors. From a comparison of the calculated with the spectroscopically measured frequencies it can be concluded, that our refined PES is a reasonable representation to the molecular S0 potential field. In particular, the m5 m2 + m6, Fermi resonance components have been satisfactorily reproduced. The remaining most prominent resonance pairs m1 m5 + m6, m1 + m2 2m5, 2m3 2m6 have also been obtained at reasonable locations (1138, 1117 cm1), (1643, 1639 cm1), and (592, 637 cm1), respectively. Of interest are the values for m6 and 2m6 of 305.20 and 637.89 cm1 contained in Table 2. As two times the m6 fundamental frequency is 610.40 cm1, the m6 manifold is highly anharmonic. The increasing value of the vibrational intervals with increasing energy can be attributed to a flattening of the potential resulting from the contribution of the positive F6666 and F3366 quartic oscillator terms. These observations lead to our suspicion that Gruebele’s [4] assignment of the intervals of 646.8 cm1 in the resolved fluorescence spectrum to overtones of m6 may indeed be correct and that vibrational anharmonicity may be responsible for the discrepancies between the various values for m6. In Table 3 the adjusted values of all quadratic, cubic and quartic force constants included in the PES of thiophosgene (Eq. (1)) are presented, as well as the optimized values of the three equilibrium parameters (two bond lengths and one interbond angle). In the Supplement to this work we have provided a C++ code for calculation of the potential energy for arbitrary input values of the displacements of all six vibrational coordinates from equilibrium. It will be our next aim, using the obtained quartic PES for thiophosgene, to perform calculations in the range of much higher excited vibrational levels in order to explore the vibrational mixing and IVR behavior, based on a realistic 6D potential field. Acknowledgement This research was supported by the National Research and Engineering Council of Canada. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.saa.2014.12.101. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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