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High-order expansion of T 2 e Jahn–Teller potential-energy surfaces in tetrahedral systems Daniel Opalka *, Wolfgang Domcke Department of Chemistry, Technische Universität München, D-85747 Garching, Germany

a r t i c l e

i n f o

Article history: Available online 8 June 2010

a b s t r a c t A high-order expansion of the three potential-energy surfaces of the T 2 e Jahn–Teller effect in tetrahedral systems is presented. It is shown that the expansion of the vibronic matrix can be obtained from two diagonal matrices determined from symmetry-invariant polynomials. The method is applied to the methane cation in its triply degenerate electronic ground state, which is known to exhibit an exceptionally strong Jahn–Teller effect. The potential-energy surfaces exhibit a highly anharmonic structure and multiple seams of intersections, rendering a high-order expansion of the potential matrix indispensable. An analytic expansion of the potential-energy surfaces up to 10th order has been ﬁtted to accurate ab initio data. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The Jahn–Teller (JT) effect is an important example of vibronic coupling in molecular and solid-state spectroscopy [1–7]. The Jahn–Teller theorem states that a molecule in a degenerate electronic state cannot be stable with respect to distortions that lower the molecular point-group symmetry. Since Jahn and Teller established the well-known theorem [1], the JT effect was the subject of many experimental and theoretical investigations. The best known example is the E e case, where the degeneracy of a doubly degenerate electronic state of E symmetry is lifted by a nuclear coordinate of the same symmetry ðeÞ. Molecules of tetrahedral and octahedral symmetry typically exhibit a variety of JT couplings between electronic states, involving nuclear coordinates of e and t2 symmetry. In particular, the degeneracy of electronic states of either E or T 2 symmetry is lifted by nuclear displacements of one of these symmetries. Examples of tetrahedral systems with a JT active ground state of E symmetry and a ﬁrst excited state of T 2 symmetry are the X þ 4 cluster cations of the ﬁfth main group ðX ¼ P; As; Sb; BiÞ which exhibit exceptionally strong Jahn–Teller couplings [8–10]. Another important example is the methane cation in its triply degenerate electronic ground state ð2 T 2 Þ which is subject to the T 2 ðt2 þ t2 þ eÞ JT effect. Several spectroscopic investigations over the past decades revealed highly complex (ro-)vibrational spectra as a result of the highly complicated topography of three coupled potential-energy (PE) surfaces [11–14]. Recently, we developed a method to construct analytic polynomial approximations of JT coupled PE surfaces which are capable of * Corresponding author. E-mail address: [email protected] (D. Opalka). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.06.011

exploiting the full permutation symmetry of tetrahedral molecules. Considering CHþ 4 as an important representative of the JT effect in YX 4 compounds, a high-order polynomial expansion of the six-dimensional T 2 ðt 2 þ t2 Þ PE surface was developed [15]. Since the T 2 t2 and T 2 e JT stabilization energies are of similar mag2 nitude in CHþ 4 , a systematic high-order expansion of the T 2 PE surface in the two-dimensional subspace of coordinates of e symmetry is the next step towards the construction of a complete PE surface of the 2 T 2 ground state of CHþ 4 . Obviously, such an expansion is also relevant for other systems with strong T 2 e JT couplings, e.g. the X þ 4 cations of the ﬁfth main group or various transition metal complexes. Both the E e and the T 2 e JT effects induce a characteristic threefold symmetry of the adiabatic PE surfaces. While the inclusion of higher-order terms in E e PE surfaces has been discussed earlier [16–18], to the best of our knowledge, no systematic treatment of T 2 e PE surfaces beyond second order in normal-mode displacements has been reported in the literature. Applying the techniques described in our previous work [15], we derive a high-order polynomial expansion of the PE surface of T 2 electronic states with respect to nuclear coordinates of e symmetry, based on a small number of generating polynomials. 2. Symmetry considerations The molecular point group T d is isomorphic to the symmetric group S4 of four identical entities. The three components of a triply degenerate electronic state of T 2 symmetry form an orthogonal basis of the T 2 irreducible representation of the T d group. Each of the 24 different matrices of the T 2 representation may be identiﬁed with one of 24 possible permutations of four identical atoms. It is important to note that spatial inversion is not a symmetry

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operation of the T d group. Consequently, molecular properties described by model functions that are invariant under any permutation of identical nuclei are not necessarily invariant under inversion of all cartesian coordinates. In tetrahedral systems, spatial inversion corresponds to an odd permutation of nuclei, followed by a rotation of the molecule in space. Nevertheless, in ﬂoppy molecules it can be indispensable to account for the permutation-inversion symmetry and there are two options: either one has to use a set of functions that are invariant under rotation and inversion, or deﬁne two or more reference geometries. In the ﬁrst approach, symmetry-adapted linear combinations (SALCs) are constructed based on internuclear distances or valence angles. It is well known, however, that these do not uniquely determine the shape of the molecule [19,20]. We adopt the second alternative for the example of the methane cation to avoid the ambiguity in the coordinates. Since the spatial inversion involves large-amplitude displacements of the e coordinates, typically via a planar structure, the coordinates are mapped to a single reference geometry before the potential is evaluated. The E irreducible representation forms a matrix group that is isomorphic to the symmetric group of three identical elements S3 , which is a subgroup of S4 . There are six different representation matrices in a basis of symmetry-adapted linear combinations (SALCs) of nuclear coordinates. The electronic state of T 2 symmetry is represented by a triple of wave functions which form a basis of SALCs for the T 2 irreducible representation in T d . Similarly, there exist SALCs of nuclear displacements as a basis of the E irreducible representation. These curvilinear displacements are illustrated in Fig. 1. In order to construct the T 2 e diabatic potential matrix, the ﬁve-dimensional E þ T 2 representation must be considered. Using SALCs as deﬁned in Ref. [15], the E þ T 2 representation is generated by the direct sum of the generating matrices of the E and T 2 representations where Gi ¼ Ei Ti i ¼ 1; 2 E1 ¼

0

1

1 1

; E2 ¼

0 1 1 0

0

0

B ; T1 ¼ @ 0 1

1 0 0 0

1

0

ð1Þ 0 0 1

1

C B C 1 A; T2 ¼ @ 0 1 0 A: 0 1 0 0

ð2Þ The representation matrices are deﬁned by the symmetry properties of the SALCs and in the present work refer to linear combinations of either nuclear displacements or electronic orbitals. The coordinates (or coefﬁcients) associated to the SALCs transform as the transposed representations in Eq. (2). In what follows, coordinates of T 2 symmetry are denoted x; y; z and the vibrational mode is represented by the displacement coordinates a; b which transform as e in T d . In contrast to the standard deﬁnition, we have chosen a basis in which the C 3 operator of the E representation is not orthogonal. The matrices E1 ; E2 are related to the conventional orthogonal rotations and reﬂections by a similarity transformation. As will become apparent later, the JT Hamiltonian has a more compact expansion in this basis (in particular, there appear no square roots in the elements of the JT potential matrix). Nuclear displacements

Fig. 1. Symmetry adapted displacements of e symmetry. The coordinates a and b specify the amplitude of the nuclear displacements shown in the left and right part of the ﬁgure, respectively.

parameterized by the coordinate vectors ðu; 0Þ; ð0; uÞ or ðu; uÞ ðu 2 RÞ with respect to a SALC basis correspond to identical distortions in different spatial directions or, equivalently, act on different nuclei. These displacements distort the tetrahedral system along the three principal axes of the reference geometry. 3. The T 2 e Hamiltonian We assume that the triply degenerate electronic ground state of CHþ 4 is isolated from other electronic states. The PE surface can therefore be constructed in the electronic subspace of the three components of the T 2 ground state. The derivation of the T 2 e Hamiltonian up to high orders is based on the formalism described in our previous work [15]. The expansion of the JT PE matrix is determined by the ﬁrst and second derivatives of the totally symmetric invariant polynomials with respect to the coordinates that transform as the coefﬁcients of the electronic T 2 state. In a ﬁrst step, a generating set of polynomials was determined which generates all invariant polynomials with respect to the ﬁve-dimensional E þ T 2 representation. Since the coordinates transform covariantly with respect to the basis functions of the representation deﬁned by the Gi in Eq. (2), the invariant polynomials in these coordinates are determined by the transposed representation generated by GT1 and GT2 . We used the software package Singular [21,22] to ﬁnd a minimal generating set of the invariant algebra in the symmetry coordinates. The complete set of polynomials generating the invariant ring R½a; b; x; y; zET 2 is given by

f 1 ¼ x2 þ y 2 þ z 2 f2 ¼ xyz f 3 ¼ x4 þ y 4 þ z 4 2

g 1 ¼ a2 ab þ b 3 2 2 3 a b þ ab þ b g 2 ¼ a3 2 h1 ¼ ða 2bÞx2 þ ða þ bÞy2 þ ðb 2aÞz2 2

ð3Þ

2

h2 ¼ a2 x2 þ ða bÞ y2 þ b z2 4

h3 ¼ ða 2bÞx þ ða þ bÞy4 þ ðb 2aÞz4 The polynomials in Eq. (3) may be used to derive any X Y; ðX; Y 2 T 2 ; EÞ JT expansion up to arbitrary order, following the procedure described in the present work. The polynomials g 1 and g 2 , which are required for the E e JT problem, are just the totally symmetric second and third order terms of Viel and Eisfeld [16] in a non-orthogonal basis. The polynomials f1 ; f 2 ; f 3 are the well-known generating invariants of the T 2 representation and deﬁne the expansion of the T 2 t2 JT matrix. The T 2 e JT matrix expansion is determined by the polynomials h1 and h2 which couple coordinates of e and t 2 symmetry. We obtained the two generating matrices

0

a 2b 0 B A¼@ 0 aþb

0 0

0

b 2a

0

1 C A;

0

a2

B B¼@ 0 0

0 ða bÞ2 0

0

1

0C A b

ð4Þ

2

for the T 2 e potential matrix from the Hessian matrix of the generating polynomials h1 and h2 in Eq. (3) with respect to the coordinates of T 2 symmetry. As shown in Ref. [15], the expansion is determined by the ﬁrst and second derivatives of the generating invariants. Since we consider here only the T 2 e problem, all x; y or z were put to zero after the matrices have been constructed and all ﬁrst derivatives vanish. It turns out that the structure of the JT expansion can be simpliﬁed if a third (redundant) generating matrix is deﬁned:

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0

1 2 b ab 0 0 1 2 B C C¼ A B ¼@ 0 ab 0 A: 4 0 0 a2 ab

ð5Þ

It is clear from the deﬁnition of C that all powers of A higher than one can be omitted, since

A2 ¼ 4C B:

ð6Þ

Furthermore, the totally symmetric polynomials g 1 and g 2 are already included by

g 1 I ¼ B þ C; 1 g2I ¼ A B C ; 2

ð7Þ

where I is the unit matrix of dimension 3. From these matrices follows an expansion which is similar to a binomial expansion in B and C. All terms of even order ð2nÞ are given by the terms in the expansion of ðB þ CÞn :

Vð2nÞ ða; bÞ ¼

n X

kk Bnk Ck ; ð2nÞ

ð8Þ

k¼0

where the kk are arbitrary ﬁtting parameters. Terms of odd degree ð2n þ 1Þ are obtained by multiplication of the term of order 2n with A:

Vð2nþ1Þ ða; bÞ ¼

n X

ð2nþ1Þ

kk

ABnk Ck :

ð9Þ

k¼0

using the correlation-consistent polarized valence triple-f basis set. The calculations were carried out with the full-valence active space. The C1s core orbital was kept closed, but also optimized in the CASSCF step. In the MRCI calculations, the core orbital was frozen. More than 1300 ab initio points with energies up to 5 eV above the energy of the T 2 state at the reference geometry have been computed with the MOLPRO quantum chemistry software [23]. For the expansion up to 10th order, 35 parameters have to be adjusted to reproduce the ab initio data as accurately as possible. Energies up to 5 eV have been taken into account in the parameter optimization. Intersections of components of the T 2 electronic state with states of different electronic character at energies below 5 eV were observed for calculations along cuts across the PE surface in the two-dimensional subspace of the e coordinates. Data points that belong to different electronic states have been removed by assigning a weight factor of zero in the ﬁtting procedure. Although the PE matrix is diagonal, a linear ﬁtting procedure would require a previous diabatization of the ab initio data due to several intersections between the diabatic PE surfaces. We rather employed the Marquardt–Levenberg algorithm to perform an iterative optimization of the parameters. For the cut along a single nuclear coordinate, we chose the coordinate a, which allowed us to exploit C 2v symmetry in the ab initio calculations (the molecule has D2d symmetry along this cut). If one of the e coordinates is zero, the parameter space in our model Hamiltonian comprises 19 parameters. Since the distortion along one of the e coordinates preserves D2d symmetry, the T 2 electronic state splits into a degenerate E state and an A state. Inspection of the expansion terms shows that the 19 parameters

In Table 1 the powers of B and C are explicitly listed which have been used in the expansion of the T 2 e JT matrix up to 10th order in the nuclear coordinates. The expansion problem is thus reduced to elementary combinatorics of binomials in the matrices of B and C. The resulting T 2 e potential is a diagonal matrix (since A, B, and C are diagonal) which is the sum of the expressions in Eqs. (8) and (9)

Vða; bÞ ¼

N X

Vn ða; bÞ:

ð10Þ

n¼1

Depending on the choice of coordinates, it can be convenient to divide the parameter space into two parts. The ﬁrst set consists of all terms Ak Bl and Ak Cl ; k ¼ 0; 1; l 2 N. Any other term of the T 2 e expansion vanishes if either a or b is zero. The remaining terms of the type Ak Bl Cm ; m 2 N, describe the coupling between the two coordinates of E symmetry. The beneﬁt of this rearrangement becomes obvious during the ﬁtting of ab initio data, where the parameters of the ﬁrst partition can be ﬁtted as a one-dimensional function to the ab initio electronic energies for displacements of a single coordinate. 4. Application to CHþ 4 We performed accurate ab initio calculations for the methane cation in the two-dimensional subspace of the nuclear coordinates a; b of e symmetry. The ab initio energies of the T 2 state were calculated with the complete-active-space self-consistent-ﬁeld/multi-reference conﬁguration-interaction (CASSCF/MRCI) method

Fig. 2. Adiabatic PE surfaces along coordinate a of e symmetry, for b ¼ 0. The circles represent energies obtained by ab initio calculations. The solid lines are the analytic adiabatic potentials of the JT Hamiltonian, expanded up to 10th order. Circles which are not connected by solid lines belong to additional electronic states (i.e., electronic states which are not components of the triply degenerate ground state at the reference geometry).

Table 1 Even-order generating polynomials deﬁned by Eqs. (3) and (5) in the T 2 e expansion up to 10th order. The ﬁrst and second order terms are A, B and C. The odd-order terms of degree 2n þ 1 are obtained by multiplying the 2n-order term with A and thus have been omitted. Tot. deg. B C

2 1 0

4 0 1

2 0

1 1

6 0 2

3 0

2 1

8 1 2

0 3

4 0

3 1

2 2

10 1 3

0 4

5 0

4 1

3 2

2 3

1 4

0 5

D. Opalka, W. Domcke / Chemical Physics Letters 494 (2010) 134–138

137

Fig. 3. Adiabatic PE surfaces for two different cuts in the subspace of e coordinates. The circles are the data points from ab initio calculations. The solid lines are the analytic adiabatic potentials of the JT Hamiltonian, expanded up to 10th order. Circles which are not connected by solid lines belong to additional electronic states (i.e., electronic states which are not components of the triply degenerate ground state at the reference geometry).

contributing to the potential in this one-dimensional subspace can be divided into sets. The parameter of the ﬁrst-order matrix in the Jahn–Teller expansion determines the gradients of the diabatic states at the reference geometry, which are related by

2

@V E ða; bÞ @a

¼ a;b¼0

@V A ða; bÞ @a

:

ð11Þ

a;b¼0

Higher-order terms of the form Ak Bl or Ak Cl ; k ¼ 0; 1; l 2 N, represent potentials of either A or E symmetry, respectively. Fig. 2 displays the corresponding data points along with the ﬁtted analytic PE surfaces. The local maximum of the lowest graph in the center corresponds to a geometry of D4h symmetry, with a planar quadratic arrangement of the hydrogen atoms. From the ratio of the group orders of T d and the epikernel subgroup D2d , it follows that there are three such planar structures (D4h is not a subgroup of T d ). For all other structures of D2d symmetry, the number of energetically identical points is twice the ratio T d =D2d due to the invariance under spatial inversion. As mentioned in the previous section, spatial inversion through a planar structure is not a symmetry operation of the molecular point group and cannot be achieved by permutations of identical nuclei. Thus in our coordinates, which depend on a space-ﬁxed reference geometry, all coordinates that belong to geometries beyond the inversion threshold have been mapped to values that describe an energetically identical structure that is close to the reference orientation. The mapping was implemented numerically as a transformation from e-type symmetry coordinates to cartesian coordinates, appropriate reﬂections in the planes deﬁned by the three planar D4h geometries of the molecule and subsequent back-transformation to symmetry coordinates. The remaining 16 parameters represent terms involving both e coordinates and have been ﬁtted to 5 different cuts across the T 2 e PE surface. The parameter optimization beneﬁts from the previously ﬁtted parameters of the potential along a single coordinate, which already provides a reasonable approximation for the PE surface for combined displacements in a and b. Fig. 3 shows two cuts of the ﬁtted PE surface. Even though they are of considerable complexity, the ﬁtted polynomial expansion (solid lines) reproduce the data points (circles) with high accuracy. It should be mentioned that displacements in the e coordinates in tetrahe-

dral systems do not completely destroy the symmetry. For arbitrary displacements, the molecule still exhibits D2 symmetry. Therefore, there exist 2T d =D2 ¼ 12 identical energy points (if spatial inversion symmetry is taken into account) which considerably simpliﬁes the acquisition of the ab initio points. The parameter optimization by the Marquardt–Levenberg algorithm was found to be straightforward.

5. Summary A high-order expansion of the PE surfaces of the T 2 e JT Hamiltonian has been developed. Methods from invariant theory have been used to obtain a complete set of polynomials which are invariant under the permutation of identical nuclei. The expansion of the diabatic PE matrix up to arbitrary polynomial degree is obtained by a simple combinatorial scheme. As an example, the method was applied to the T 2 e JT effect in the methane cation in its electronic ground state. The analytic PE functions have been ﬁtted to ab initio data up to an energy of 5 eV above the energy at the reference geometry. Expansion terms complete up to polynomials of 10th order have been considered. Despite the large JT stabilization energy of the e mode (1.3 eV) and the extreme anharmonicity of the T 2 e PE surface, an accurate analytic representation of the ab initio data points could be obtained over a wide range of the displacement coordinates of e symmetry. The results demonstrate that high-order polynomial expansions are very useful for accurate representations of PE surfaces in cases of strong JT coupling. The described method is suitable for of any combination of T 2 or E electronic states, coupled to nuclear coordinates of t2 or e symmetry. The expansion of the JT PE surface can be extended to an arbitrary number of t2 or e nuclear coordinates with the polarization procedure described in Refs. [15,24]. This is important, for example, in YX 3 or YX 4 systems, in which two nuclear coordinates of e or t 2 symmetry, respectively, exist. A high-order expansion of the T 2 ðt 2 þ t 2 Þ JT PE surface of CHþ 4 has been obtained in Ref. [15]. An accurate T 2 e JT PE surface has been generated in the present work. The ﬁnal T 2 ðt2 þ t2 þ eÞ JT PE surface of the ground state of CHþ 4 is under construction.

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