The Quantization of the E â e JahnâTeller Hamiltonian - The Journal

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The Quantization of the E × e Jahn-Teller Hamiltonian Athanasios G. Arvanitidis, Eva R.J. Vandaele, Marek Szopa, and Arnout Ceulemans J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b06136 • Publication Date (Web): 29 Aug 2017 Downloaded from http://pubs.acs.org on September 2, 2017

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The Journal of Physical Chemistry

The Quantization of the E ⊗ e Jahn-Teller Hamiltonian Athanasios G. Arvanitidis,† Eva R. J. Vandaele,† Marek Szopa,‡ and Arnout Ceulemans∗,† †Department of Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium ´ aski, Uniwersytecka 4, PL-40-007 Katowice, Poland ‡Uniwersytet Sl¸ E-mail: [email protected] Abstract The E ⊗ e Jahn-Teller Hamiltonian in the Bargmann-Fock representation gives rise to a system of two coupled first-order differential equations in the complex field, which may be rewritten in the Birkhoff standard form. General leapfrog recurrence relations are derived, from which the quantized solutions of these equations can be obtained. The results are compared to the analogous quantization scheme for the Rabi Hamiltonian.

Introduction The E ⊗ e Jahn-Teller (JT) Hamiltonian describes the coupling of a two-level fermion system with two degenerate vibrational modes. The simple linear coupling case can be seen as the offspring of an even more fundamental interaction model of a similar two-level fermion system with a single bosonic mode, described by the Rabi Hamiltonian. In fact the rotational symmetry of the famous Mexican hat potential of the linear JT system reduces the 1 ACS Paragon Plus Environment


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coordinate space to an effective single radial mode. However while in the Rabi model the coupling constant between the two fermion states is constant, in the JT case it is equal to the centrifugal term, implying that it is a function of the radial distance to the center. It was shown previously how a sequence of transformations made it possible to reduce both Hamiltonians to the same canonical form. 1 Recent developments which have led to new insights into the quantization structure of the Rabi Hamiltonian, 2,3 have prompted us to revisit the JT case, with particular attention to its quantization structure.

The model system The model system consists of two fermion states, coupled by a linear force element to a twofold degenerate harmonic oscillator. In octahedral symmetry, the two real components of the fermion manifold are typically denoted as |θi and |i ket functions, and the modes are p generated by creation operators a†θ and a† . A unit of length is defined as ~/mω and the oscillator quantum ~ω is taken as the unit of energy. This rescaling absorbs all fundamental constants, e.g.: 1 a†θ = √ (Qθ − iPθ ) 2

(1) 



 |θi  The Hamiltonian can be written in a matrix form, acting in the space  : |i   H=

a†θ aθ

+

a† a

+1+λ

−λ

a†

+ a

a†θ

+ aθ

−λ

a†



+ a   † † aθ a + a† a + 1 − λ aθ + aθ

(2)

Here, λ is the linear coupling parameter, and 1 is the zero-point energy. Subsequently the boson vacuum state is shifted to the zero of energy. A zero-field potential splitting between the two fermion states, as in the Rabi case, can be included as well, but here we will concentrate on the simplest case.

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In the vast literature on the Jahn-Teller effect 4–6 coupling parameters and phases have been defined in several ways. Usually the linear coupling element, which represents the distortion force, is denoted by a coupling constant, FE , which relates to our λ, as: √ FE = −λ 2

(3)

Such is the case in the seminal treatment of the dynamical problem by Longuet-Higgins. 7 In the later analytical treatment by Reik and Doucha the coupling parameter was denoted κ. 8 It follows the same relationship but with opposite phase: √ κ=λ 2

(4)

The present λ parametrization was adopted because it allows us to establish a close relationship between the Jahn-Teller and Rabi Hamiltonian. 1,9 The deformation energy is expressed as λ2 . This is equivalent to the coupling strength, D, introduced by Ballhausen: 6

D = λ2

(5)

Rotational symmetry The JT potential has an obvious rotational symmetry, so one can define an angular momentum operator in the space of the vibrational coordinates. We will denote it as:

ˆ z = Qθ P − Q Pθ L i † i h † † † a + aθ a − a − a + a aθ − aθ = 2 θ = i a† aθ − a†θ a


(6)


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However a straightforward calculation of the commutator of this operator with the Hamiltonian reveals that they do not commute! One has: 

 h i + a λ + aθ  λ ˆz, H = i  L   λ a†θ + aθ −λ a† + a a†

a†θ

(7)

Indeed symmetry should not be restricted to the boson part only, but involves both the bosons and the fermions. So we have also to take into account the changes of the fermion states when we revolve along the trough of the Mexican hat surface. Along the lower sheet of the potential energy surface the composition of the adiabatic wavefunction rotates in the space of the fermion components, but at half speed and in opposite sense as compared to the distortion coordinate. Hence the corresponding operator can be thought of as a spin ˆ z . It is thus defined as follows: momentum operator, associated with L i Sˆz = (|θih| − |ihθ|) 2

(8)

h i h i ˆ ˆ Sz , H = − Lz , H

(9)

One easily verifies:

The total angular momentum operator will then be represented as Jˆz : ˆ z + Sˆz Jˆz = L

(10)

This operator generates the SO(2) rotational symmetry of the JT system. Having identified the algebra, the first task now will be to perform a symmetry adaptation of the boson


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operators and of the fermion states. For the boson operators one has: 1 a†+ = √ a†θ + ia† 2 1 a+ = √ (aθ − ia ) 2 1 † † a− = √ aθ − ia† 2 1 a− = √ (aθ + ia ) 2

(11)

ˆ z with opposite eigenvalues: These operators are eigenoperators of L h

ˆ z , a†± L

i

= ±a†±

h

ˆ z , a± L

i

= ∓a±

(12)

The corresponding states are given by: 1 † n a± |0i |ni± = √ n!

(13)

These states are eigenfunctions of the momentum operator:

ˆ z |ni± = ±n|ni± L

(14)

ˆ z |ni+ |mi− = (n − m)|ni+ |mi− L

(15)

For a combined excitation one has:

Hence to realize an excitation with Lz value equal to l > 0, n should outnumber m by the amount l, with: n = m + l. Similarly one should recombine the fermion state to complex



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conjugate forms by: 1 | ↑i = √ (|θi − i|i) 2 1 | ↓i = √ (|θi + i|i) 2

(16)

As eigenfunctions of the Sˆz operator, these combinations are analogous to α and β spins:

1 Sˆz | ↑i = + | ↑i 2 1 Sˆz | ↓i = − | ↓i 2 The  total symmetry-adapted Hamiltonian is now expressed in   | ↑i  basis : | ↓i  √ † † † a a + a a λ 2 a + a + − + + − −  H= √ a†+ a+ + a†− a− λ 2 a†+ + a−

(17)

the transformed fermion

  

(18)

The Canonical Form of the wave equation In the standard Schrödinger approach the wavefunction is expanded in the basis space of the phonon and fermion states, and a matrix equation is derived from which the expansion coefficients may be obtained. The work of Reik et al. 8,10 presents an alternative treatment which is based on the Bargmann-Fock mapping. 11 This is also the method that we will use here. The boson creation and annihilation operators are mapped onto complex variables and their derivatives as follows: a†+ → ξ1

a+ →

∂ ∂ξ1

a†− → ξ2

a− →

∂ ∂ξ2


(19)

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The Ansatz for a wavefunction with angular momentum j = l +

1 2

is then given by:

|Ψil+1/2 = (ξ1 )l Φ1 (ξ)| ↑i + (ξ1 )l+1 Φ2 (ξ)| ↓i,

(20)

with ξ = ξ1 ξ2 . Note that, because of time-reversal symmetry, each wavefunction of this type forms a degenerate doublet with a complementary wavefunction with opposite momentum −j, given by: |Ψi−l−1/2 = (ξ2 )l Φ1 (ξ)| ↓i + (ξ2 )l+1 Φ2 (ξ)| ↑i.

(21)

The Hamiltonian is now applied to the wavefunction. This results in two coupled first-order differential equations, which determine the eigenstates of the two fermion components. The eigenenergy is denoted as E, where we have incorporated the zero-point energy. √ ∂ ∂ 0 = (l − E)Φ1 + 2ξ Φ1 + λ 2 ξ + (l + 1) + ξ Φ2 ∂ξ ∂ξ √ ∂ ∂ 0 = (l + 1 − E)Φ2 + 2ξ Φ2 + λ 2 1 + Φ1 ∂ξ ∂ξ

(22)

√ This result was already obtained by Reik et al., with κ = λ 2. 8 Note that the individual variables ξ1 and ξ2 only appear as their product ξ. The symmetry adaptation to the rotation group thus has effectively reduced the problem to a single dimension. Since the Φ1 and Φ2 functions only depend on ξ the following identities are implied:

ξ1

∂ ∂ ∂ = ξ2 =ξ ∂ξ1 ∂ξ2 ∂ξ

(23)

Following the earlier result by Szopa and Ceulemans 1 we now continue by symmetrizing √ Eq.(22). Since ξ1 ξ2 has zero angular momentum, we can think of ξ as a ’radial’ creation operator. In the Ansatz the spin-up and spin-down part are not of the same rank in this variable. We thus introduce two new functions, f1 and f2 , where this difference of rank is



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removed: p l ξ Φ1 p l+1 ξ Φ2 =

f1 = f2

(24)

Carrying out this substitution in Eq.(22), and taking out common factors, yield a modified set of equations: √ p ∂ l+1 p ∂ −Ef1 + 2ξ f1 + λ 2 ξ+ √ + ξ f2 = 0 ∂ξ ∂ξ 2 ξ p ∂ √ p ∂ l −Ef2 + 2ξ f2 + λ 2 ξ− √ + ξ f1 = 0 ∂ξ ∂ξ 2 ξ

(25)

We now introduce symmetric and antisymmetric combinations, and introduce a proper radial √ variable z ≡ 2ξ: f1 + f2 2 f1 − f2 Ψ2 (z) = 2 Ψ1 (z) =

The symmetry of the f -functions with respect to z is determined by the powers of

(26) √

ξ, since

Φ1 and Φ2 only depend on z 2 . Hence: f1 (−z) = (−1)l f1 (z) f2 (−z) = (−1)l+1 f2 (z)

(27)

Accordingly the symmetry of the Ψ-functions with respect to a sign change of z is obtained as:

Ψ1 (−z) = (−1)l Ψ2 (z) Ψ2 (−z) = (−1)l Ψ1 (z) 8 ACS Paragon Plus Environment

(28)

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In this way we obtain a symmetrical system of coupled first-order differential equations: 1 −2 E + λ2 + 12 d 1 1 1 Ψ1 = + − λ Ψ1 + l + − Ψ2 dz z z+λ 2 z z+λ 1 −2 E + λ2 + 21 1 1 1 d Ψ2 = l+ − Ψ1 + + + λ Ψ2 dz 2 z z−λ z z−λ

(29)

The solutions Ψ1 and Ψ2 together with the appropriate energy E represent physical states provided the corresponding Φ1 and Φ2 belong to the Bargmann-Fock space i.e. are entire and obey the appropriate normalizability condition. 9 Note, that these equations are invariant under a sign change of z, when concomitant with an interchange of Ψ1 and Ψ2 . As expressed in Eq.(28), the eigenstates have a fixed parity under this interchange, which depends on the angular momentum quantum number l. This is reminiscent of the parity of the states in the Rabi Hamiltonian. We will come back to this point in the discussion. The set of equations has three finite singular points at z = 0, ±λ. Following our earlier treatment we now apply the Birkhoff transformation, 12 which yields a canonical form with only one singularity in the origin. Let us first rewrite the equations in a more general way as: d Ψ1 = p11 (z)Ψ1 + p12 (z)Ψ2 dz d Ψ2 = p21 (z)Ψ1 + p22 (z)Ψ2 dz

(30)

or, d Ψ = pΨ dz

(31)

Outside the circle |z| = λ the pij -coefficients can be expanded in a Laurent series

pij =

q X

(k)

(k)

pij z k , pij ∈ C

(32)

k=−∞

Here q + 1 is the rank of the singular point at infinity, called the Poincaré rank of Eq.(31).



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For the Jahn-Teller Hamiltonian q = 0 hence the rank is equal to 1. Now we assume a linear transformation of the form: Ψ = aF

(33)

where the transformation coefficients aij (z) are analytic at infinity and reduce at infinity to the unit matrix: aij (z) =

(k) ∞ X aij k=0

zk

(k)

, aij ∈ C

(34)

This matrix transformation contains all the finite singularities of the initial system. In view of the symmetry of this matrix we adopt a simplified notation as:

a11 (z) = a12 (z) =

∞ X ak k=0 ∞ X k=0

zk bk zk

a21 (z) = a12 (−z) a22 (z) = a11 (−z)

(35)

with: a0 = 1 and b0 = 0. By combining these expressions the original set of equations can be turned into a transformed system:

z

d F = PF dz

(36)

If the matrix P(z) is a polynomial of degree equal to the Poincaré rank, then Eq.(36) is the canonical form of Eq.(31), called the Birkhoff standard form. 13 In the original paper 12 Birkhoff falsely claimed that each system (31) can always be reduced to the standard form. Later it was shown (see e.g. 13 ) that such a transformation is possible provided the initial system is irreducible. A system of Eq.(31) is called reducible if an analytic transformation (33) exists for which the transformed equation is (lower) triangularly blocked. The sufficient (q)

condition for the irreducibility of the initial system is that the characteristic roots of {pij } 10 ACS Paragon Plus Environment

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(0)

are distinct. 14 In our case pij = (−1)i δij λ, therefore the Eq.(31) indeed can be reduced to the Birkhoff standard form. The coefficients of the standard form are related to the original coefficients by the following matrix transformation: d 1 aP = pa − a, z dz

(37)

where the Pij coefficients are polynomials of a degree that does not exceed the Poincaré rank which, in case of Eq.(29), is 1. They can thus be easily obtained from the previous equation by collecting the terms in 1/z k with k = 0, 1. The Birkhoff standard form of the Jahn-Teller equation thus reads: d F1 = E + λ2 − λz F1 − 2λb1 F2 dz d z F2 = −2λb1 F1 + E + λ2 + λz F2 dz z

(38)

In addition, in view of Eq.(34), the original and transformed system are required to share the same reflection symmetry: 







F2 (−z)   F1 (z)  l   = (−1)   F2 (z) F1 (−z)

(39)

In Eq.(38) the factor 2λb1 will be called the gauge factor, A, as will be explained later on.

A = 2λb1

(40)

If one includes in the Hamiltonian a level splitting, 2∆, at the coordinate origin, the gauge becomes ∆ + 2λb1 . As we have noted before 1 the resulting set of differential equations is in this case identical to the Birkhoff standard form of the Rabi Hamiltonian. The solutions are thus the same as well, but the recurrence relations are different. By eliminating F2 the



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canonical set may be transformed into a second-order differential equation in F1 : z 2 F100 (z) + z[1 − 2(E + g 2 )]F10 (z) + [(E + g 2 )2 − A2 + gz − g 2 z 2 ]F1 = 0

(41)

The roots of the indicial equation read:

ρ± = E + λ2 ± A

(42)

The differential equation can be reduced to the Kummer equation, which is solved by the confluent hypergeometric functions 1 F1 (a, c; z). 15 The general solution reads: F1 (z) = C1 exp(λz) 1 F1 (1 + A, 1 + 2A; −2λz) z E+λ

2 +A

(43) + C2 exp(λz) 1 F1 (1 − A, 1 − 2A; −2λz) z

E+λ2 −A

By inserting the solution for F1 in Eq.(38) we obtain the F2 component: F2 (z) = −C1 exp(−λz) 1 F1 (1 + A, 1 + 2A; 2λz) z E+λ

2 +A

(44) + C2 exp(−λz) 1 F1 (1 − A, 1 − 2A; 2λz) z

E+λ2 −A

We are now in a position to determine conditions under which the above solutions belong to the Bargman-Fock space. The Kummer functions are entire therefore both solutions are entire provided at least one of the roots of the indicial equation is a non-negative integer. The asymptotic behaviour of the solutions is determined by the factors exp(±λz), which implies that they obey the normalizability condition. The C1 and C2 solutions give rise to a different parity expression. For the C1 solution one has:









F2 (−z)   F1 (z)  ρ +1    = (−1) +   F2 (z) F1 (−z)


(45)

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For the C2 solution the parity is opposite and is given by: 







F2 (−z)   F1 (z)  ρ   = (−1) −    F1 (−z) F2 (z)

(46)

According to Eq.(39), the parity of the solution is dictated by the parity of the angular quantum number l. Hence one must carefully choose the parity of the roots: for the ρ+ root, ρ+ and l should have opposite parity, while for the ρ− root, they should have the same parity. In the next paragraph we explore the relation in Eq.(37) to find conditions under which the transformation matrices have a finite expansion. It is a sufficient condition also for the solutions of the initial system Φ1 and Φ2 to belong to the Bargmann-Fock space. The corresponding energy values will be therefore the eigenvalues of the Jahn-Teller problem.

Recurrence relationships The ak and bk coefficients are determined by recurrence relations. As for the Rabi case we have been able to work out a general formalism for these relations, based on the general expression in Eq.(37), by grouping together terms in the same power of z. For z → ∞ the transformation matrix must approach the unit matrix, hence the recurrence series starts off at:

a0 = 1 b0 = 0

(47)

The b1 parameter is unique in that it is featuring in the gauge parameter. By requiring one root of the indicial equation to be a non-negative integer, say ρ+ = k, the b1 parameter is



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expressed as: b1 = (k − E − λ2 )/(2λ)

(48)

From here on the recursive expressions take a start, generating one coefficient at the time, in a leapfrog sequence as: b1 → a1 → b2 → a2 → b3 .... 1 a1 = −2λb21 + (E + λ2 + )λ 2 1 b2 = b1 + 2λb1 a1 + (l + )λ /(2λ) 2 1 1 2 2 a2 = −2λb1 b2 + (E + λ + )(λa1 − λ ) + (l + )λb1 /2 2 2 1 1 2 2 b3 = 2b2 + 2λb1 a2 − (E + λ + )λb1 − (l + )(λ + λa1 ) /(2λ) 2 2 1 1 2 3 2 2 a3 = −2λb1 b3 + (E + λ + )(λa2 − λ a1 + λ ) − (l + )(λb2 + λ b1 ) /3 2 2 (49)

The resulting expressions can be recast in general as two consecutive recurrence formulae which are valid from a2 and b3 onwards: n

n−1

1 X ν 1 X (−1)ν+1 λν an−ν + (−1)n (l + ) λ bn−ν nan = −2λb1 bn + (E + λ2 + ) 2 ν=1 2 ν=1 2λbn

n−1 n−2 1 X ν 1 X 2 = (n − 1) bn−1 + 2λb1 an−1 + (−1) (l + ) λ an−ν−1 + (E + λ + ) (−1)ν λν bn−ν−1 2 ν=1 2 ν=1 n

(50)

By combining nan + λ(n − 1)an−1 one then obtains the first four-term recursion relation, which generates an from an−1 and bn , bn − 1 coefficients:

1 1 n nan = E + g + − (n − 1) λan−1 − 2λb1 bn + (l + )(−1) − 2λb1 λbn−1 2 2 2

(51)

Similarly by combining 2λbn +2λ2 bn−1 one obtains the second five-term recursion relation,


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which generates bn from the previous an−1 , an−2 , bn−1 , and bn−2 coefficients. 1 n 2λbn =λ (l + )(−1) + 2λb1 an−2 + 2λb1 an−1 2 1 2 + λ n − 2 − (E + λ + ) bn−2 + (n − 1 − 2λ2 )bn−1 2

(52)

These recursions are very similar to the ones from the Rabi Hamiltonian. We will elaborate this point further in the discussion.

Results By working out the recursion relations, all coefficients are expressed as increasing polynomials of the energy. The eigenenergies are approximated as the roots of these polynomials. All calculations and graphics were performed with the Octave software. 16 In Figure 1 we show some polynomials of the bn coefficient for increasing values of n. It is observed that already for these low values of n the roots coincide. The curves become more pronounced for increasing n and the intersection with the zero-line becomes sharper and sharper, indicating that more accurate roots can be obtained by truncating the series at higher values of n. This is illustrated in Figure 2 where we plot the polynomials for a different range with higher n values. In Table 1 we list the lower roots for l = 0, using the zeroes of the b30 polynomial, and list for comparison the ’historical’ values obtained by Longuet-Higgins et al. 7 using matrix diagonalization on one of the first digital machines in the UK in 1953. Repeating this procedure with higher accuracy shows that diagonalization of a 30 × 30 matrix already reproduces the lower roots up to five digits.

As was already mentioned, in addition to

choosing bn one must also decide on the value of the quantum number k. The parity of k depends on the quantization condition which is imposed. If one adopts k = ρ− , k should have the same parity as l. Hence if we are looking for roots with l = 0, 2, 4, ... and make use of k = ρ− we should adopt even values of k. Roots for odd values of l should be combined with odd values of k. The opposite is true if one adopts k = ρ+ . In this case, the roots for 15 ACS Paragon Plus Environment


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bn

200

100

n=6 n=7 n=8

0

-100

-200

-1

-0.5

0

0.5

1

1.5

2

2.5

E

Figure 1: Plot of the transcendental function bn (E + λ2 ) for n = 6 (blue), n = 7 (green), n = 8 (red). The k-value is set to zero, and the ρ− root is chosen. The roots correspond to symmetric eigenfunctions with l = 0.

bn 1e+14

5e+13

n=18 n=19 n=20

-5e+130

-1e+14

-1

0

1

2

3

4

E

Figure 2: Plot of the transcendental function bn (E + λ2 ) for n = 18 (blue), n = 19 (green), n = 20 (red). The k-value is set to zero, and the ρ− root is chosen. The roots correspond to symmetric eigenfunctions with l = 0.


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√ Table 1: Eigenenergies for λ = 1/ 2, l = 0, using ρ− = 0, with b30 = 0. The right column provides the data in Ref. 7 The boson vacuum energy is taken as zero energy. Ref. 7 −0.767 +0.488 +1.569 +2.435 +3.536 +4.493 +5.485 +6.529 +7.467

This work −0.766969 +0.487794 +1.569454 +2.434572 +3.535933 +4.492497 +5.484479 +6.529441 +7.466831

even l are obtained by inserting odd values of k and vice-versa. Since k is an integer, it can fulfill the role of a quantum number. In principle the proper k value of an eigenfunction is the one for which the expansion converges fastest. This implies that k can be used as a kind of slide ruler which is adapted to the spectral region. As an example in Table 2 we compare accurate values for the strong coupling case, for different k and n. The higher roots are calculated more accurately by adopting higher k values. In the final column of Table 2 we compare the results to the eigenvalues of a 30 × 30 matrix diagonalization. Increasing the matrix dimension to 100×100 yields exactly the same eigenvalues as the b100 = 0 run. Figure 3 shows the spectrum as a function of the coupling constant for l = 0, 1, 2, 3. The overall energy asymptotically decreases linearly with λ2 . We thus have plotted E + λ2 as a function of λ2 , yielding horizontal asymptotes. As for the Rabi case the spectrum was also found to include non-physical roots. In the diagram we have indicated the highest unphysical root for each l value. These roots are almost linearly increasing with λ2 as opposed to the physical eigenstates.

Discussion With increasing coupling strength, the evolution of the E ⊗ e Jahn-Teller eigenvalues reflects the transition from a two-dimensional harmonic oscillator, to a one-dimensional radial os17 ACS Paragon Plus Environment


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Figure 3: Spectrum of the Jahn-Teller Hamiltonian as a function of the coupling strength λ2 . The eigenenergies are plotted as E + λ2 . The dashed lines with positive slopes represent the unphysical solutions, characterized by increasing values of l, the lowest one corresponding to l = 0.

Table 2: Eigenenergies for strong coupling, with λ = 2.5, and l = 0, using ρ− = k, for different bn and k values. Comparison to the variational method for a 30 × 30 matrix. k=0 b30 = 0 −6.738349 −5.735879 −4.723221 −3.716224 −2.678227 −1.626603 −0.538010 +0.580516

k=0 b100 = 0 −6.738516 −5.734462 −4.727054 −3.712024 −2.681483 −1.625646 −0.537902 +0.580515

k = 12 b30 = 0 −6.738516 −5.734461 −4.727054 −3.712024 −2.681480 −1.625665 −0.537948 +0.580531


30 × 30 −6.738516 −5.734461 −4.727054 −3.712019 −2.681397 −1.624652 −0.529923 +0.623330

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cillator combined with a rotor. The states of the uncoupled 2D oscillator form a ladder of equidistant eigenenergies with integer eigenvalues starting from E = 0. The vacuum state is non-degenerate while each excitation increases the degeneracy by one. The angular composition of the uncoupled oscillator levels can be derived from Eq.(22) with λ = 0. One has: ∂ Φ1 ∂ξ ∂ 0 = (l + 1 − E)Φ2 + 2ξ Φ2 ∂ξ 0 = (l − E)Φ1 + 2ξ

(53)

Solutions of these equations can be recast as:

Φ1 = z E−l Φ2 = z E−l−1

(54)

Since z is a complex variable the requirement that the solutions be single-valued implies that they must be entire, hence the powers of z must be nonnegative integers:

E − l = 0, 1, 2, 3, ... E − l − 1 = 0, 1, 2, 3, ...

(55)

For the vacuum level at E = 0 there is only one solution with l = 0 and Φ2 = 0. So the lowest oscillator level will be characterized by l = 0. For the first excited level, at E = 1, there are two solutions, with l = 0, 1. Similarly with each new excitation a further angular component is added, hence the state with E = 3 is fourfold degenerate and includes l = 0, 1, 2, 3, etc. When the Φ solutions belong to the Bargmann-Fock space of physical states, the Birkhofftransformed Ψ functions will also be entire and belong to this space. However the reverse is not necessarily true, and this explains the appearance of ’unphysical’ solutions. Indeed for



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λ = 0 the Birkhoff equations reduce to: d F1 = EF1 dz d z F2 = EF2 , dz z

(56)

yielding F1,2 = z E . Here the value of l no longer appears, so for every integer value of E one finds states with l-values that are in fact excluded by the physical requirements, simply because they are not entire. In the diagram we have included for each l the highest level which is not entire. For l = 0 this state appears at E = −1, for l = 1 it corresponds to E = 0 etc. It is remarkable that these states show a consistent rise in the diagram. Special solutions in the spectrum are related to the Kummer functions 1 F1 (a, c; z) that solve the Birkhoff transformed differential equations. As indicated in Eq.(43) the c parameter of these functions is given by: c = 1 ± 2A

(57)

When c is a positive integer the Kummer series will terminate. This means that we may expect special solutions when the gauge parameter A is integer or half-integer. The half-integer values of A imply half-integer values of E + λ2 . Such values correspond to the celebrated Juddian baselines: E + λ2 + 1/2 = ν

(58)

In our previous paper 1 we have already indicated that the Birkhoff method indeed can reproduce the crossing points with these baselines which yield the Juddian exact solutions. At present we are most interested in the integer values of A. These imply integer values of E + λ2 , giving rise to intermediate baselines:

E + λ2 = ν


(59)

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Combined with the root expression for the indicial equation, this reads:

k = ρ± = E + λ2 ± A = ν ± A

(60)

Hence when one approaches such a baseline, and fixes a value of k equal to ν, then the A gauge will approach zero. In the case of the Rabi Hamiltonian the crossing always implies simultaneous crossings of symmetric and antisymmetric states in opposite directions. In the Jahn-Teller case multiple crossings are observed due to different values of l. Quite remarkably, these crossings seem to coalesce in nodal points at regular intervals on the baselines. Moreover also the unphysical solutions apparently make use of these nodal points to cross the baselines. However in the early study by Thorson and Moffitt on the analogous Γ8 ⊗ t2 Hamiltonian, it was already established by a very careful and extensive numerical analysis that the crossings are in fact ’near misses’. 17 This was further analysed by Judd who corroborated the finding of Thorson and Moffitt on the basis of a perturbational expansion in the neighborhood of the crossing. 18 Judd also showed that the distribution of the intersections on the baseline approximately corresponds to the zeroes of Besselfunctions. Our present recurrence relations provide a simple tool for the accurate determination of these zeroes. By inserting a resonant energy, E + λ2 = ν, and zero gauge, A = 0, for a fixed l-value, the recurrence relations become polynomials in λ. The real positive roots of these polynomials describe the location of the crossing points. As an example, in Table 3 we list the lowest roots of these polynomials for E + λ2 = 2 and l = 0, 1. The Table illustrates clearly that these roots can accurately be predicted by the zeroes of the expansion coefficients, already at low order. Moreover it is noticed that the crossing points are really near misses as the recurrence relations are dependent on l. As an example for b6 = 0 one obtains the following



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quartic equations in λ2 .

l=0

2169λ8 − 10256λ6 + 19372λ4 − 14400λ2 + 2880 = 0

l=1

2169λ8 − 10704λ6 + 19788λ4 − 14400λ2 + 2880 = 0

(61)

These predict crossing points at 0.555393 and 0.560195 for l = 0 and l = 1 respectively. Very precise calculations with b30 = 0 yield 0.55387958 for l = 0, and 0.560203212 for l = 1. These values illustrate the presence of an almost coalescing crossing point, as can be seen from Figure 3. Note that these polynomials determine also further crossing points, which however involve also unphysical eigenvalues. As an example the diagram in Figure 3 identifies a further near miss on the E + λ2 = 2 baseline at λ2 between the l = 0 level and the unphysical l = 1 root. Table 3: Nodal points of the physical eigenvectors on the E + λ2 = 2 baseline for l = 0, 1, as determined from the zeroes of the expansion coefficient as a function of λ. Note the near coalescence of the first roots with different l-values. n b4 b5 b6 b7 b8 b9 a4 a5 a6 a7 a8 a9

l=0 λ1 0.557213 0.555318 0.555393 0.555388 0.555388 0.555388 0.521740 0.552826 0.555092 0.555369 0.555386 0.555388

λ2 0.94806 − 1.04786 1.06702 1.06171 1.06282 − 1.22319 − 1.08332 1.07179 1.06414

l=1 λ1 0.557213 0.560506 0.560195 0.560204 0.560203 0.560203 0.544372 0.555802 0.560067 0.560185 0.560202 0.560203

Clearly the parity of the Jahn-Teller eigenstates plays an important role in the spectrum. In the case of the Rabi Hamiltonian a similar observation was made with regard to parity derived from the mirror symmetry between the two displaced oscillator wells. In the case of the Mexican hat potential the mirror relations between points on opposite sides of the 22 ACS Paragon Plus Environment

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potential energy surface is brought about by a simultaneous reflection of boson and fermion parts. Let π ˆ represent this operation. It corresponds to: π ˆ (a†θ ) = − a†θ π ˆ (aθ ) = − aθ π ˆ (a† ) = − a† π ˆ (a ) = − a π ˆ |θi = |i π ˆ |i = −|i

(62)

This is nothing other than a rotation over an angle π in coordinate space, and a rotation over π/2 in the associated space of the fermion states. The starting Hamiltonian in Eq.(2) is clearly invariant under such an operation. The symmetry adapted Hamiltonian in Eq.(18) is likewise invariant. In this case the action of π ˆ on the spinor components is as follows:

π ˆ | ↑i = i| ↑i π ˆ | ↓i = −i| ↓i

(63)

Note the appearance of the imaginary in this case. As a result applying the reflection twice will multiply the kets by a phase factor of −1. This simply reflects the presence of the Berry phase, which accompanies a full rotation in coordinate space. We are now in position to evaluate the effect of the parity operation on the total wavefunction of the Bargmann mapping. Note that the complex variables ξ1 and ξ2 will simply change sign upon reflection,



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while their product ξ will be invariant.

π ˆ |Ψil+1/2 = i (−ξ1 )l Φ1 (ξ)| ↑i − i (−ξ1 )l+1 Φ2 (ξ)| ↓i h i = i(−1)l (ξ1 )l Φ1 (ξ)| ↑i + (ξ1 )l+1 Φ2 (ξ)| ↓i = i(−1)l |Ψil+1/2

(64)

This result confirms that the parity indeed depends on the parity of the boson angular momentum l, as was established by the solutions of the Birkhoff equations. Perhaps the most intriguing aspect resulting from the present treatment, is the close analogy between the Jahn-Teller and Rabi couplings. As we already indicated, in the presence of a constant level splitting the Birkhoff forms of the two cases are identical. If we take a closer at the recurrence relations in Eq.(49), the following correspondences are noted: • The term ∆ + 2λb1 is replaced by 2λb1 as we did not consider the level splitting. • The Rabi splitting parameter ∆ is replaced in the final term of both recursions by the angular momentum l + 21 . • The energy appears in the Rabi Hamiltonian as E + λ2 , while in the Jahn-Teller case it is E + λ2 + 21 . The extra fraction

1 2

in the energy expression simply represents the increase of the zero-point

energy for the two-dimensional Jahn-Teller case as compared to the one-dimensional Rabi Hamiltonian. The replacement of the Rabi coupling parameter by the total vibronic angular momentum term, l + 12 , provides a clear physical insight into the analogy. In the Jahn-Teller case the communication between the fermion states is due to the centrifugal potential. While this term depends on the inverse of the radius squared, 6 the transformation shows that it in fact reduced to a level splitting constant, exactly equivalent to the level splitting in the Rabi case. This indicates that the gyrating molecule is in a fixed rotational state, thus giving rise to a constant dynamical coupling between the fermion states. When the orbital momentum 24 ACS Paragon Plus Environment

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of the nuclear pseudorotation vanishes (l=0), the fermionic spinor continues to serve as a coupling term.

Conclusion In 1997 Szopa and Ceulemans derived the Birkhoff standard form for the E × e JahnTeller Hamiltonian, and showed that this was identical to the Rabi case. 1 At that time this form was shown to reproduce the Juddian exact solutions, but no further results could be obtained. Recently, full solutions of the Rabi were achieved by requiring convergence of the appropriate recurrence relations. In the present paper this result was extended to the Jahn-Teller case. The rotational angular momentum of the Jahn-Teller Hamiltonian assumes the role of the splitting parameter in the Rabi case. The new insight, resulting form the transformation of the problem to a canonical form, is the appearance of a gauge potential which measures the distance between the actual eigenenergy, and the baselines corresponding to a displaced oscillator. It associates the eigenenergies to the ’effective’ quantum numbers, which characterize the baselines. Special solutions appear when this gauge potential equals integer or half-integer units of energy.

Acknowledgement A.G.A thanks the Flemish Science Fund (FWO) for financial support, and is indebted to Annelies Postelmans (MeBioS, KULeuven) for an introduction to the Octave Software.

References (1) Szopa, M.; Ceulemans, A. The canonical form of the E ⊗ e Jahn-Teller Hamiltonian. J. Phys. A: Math. Gen. 1997, 30, 1295–1302. (2) Braak, D. Integrability of the Rabi model. Phys. Rev. Lett. 2011, 107, 100401. 25 ACS Paragon Plus Environment


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(3) Vandaele, E. R. J.; Arvanitidis, A. G.; Ceulemans, A. The quantization of the Rabi Hamiltonian. J. Phys. A: Math. Gen. 2017, 50, 114002. (4) Bersuker, I. B.; Polinger, V. Z. Vibronic interactions in molecules and crystals; Springer: Berlin, 1989. (5) Judd, B. R. The Jahn-Teller effect for degenerate states. Colloques internationaux CNRS 1976, 255, 127–132. (6) Ballhausen, C. J. In in: Vibronic processes in inoganic chemistry; Flint, C. D., Ed.; Kluwer: Dordrecht, 1989; pp 53–78. ¨ (7) Longuet-Higgins, H. C.; Opik, U.; Pryce, M. H. L.; Sack, R. A. Studies of the Jahn-Teller effect II. The dynamical problem. Proc. Roy. Soc. London. A Math. Phys. Sciences. 1958, 244, 1–16. (8) Reik, H. G.; Lais, P.; St¨ utzle, M.; Doucha, M. Exact solution of the E ⊗ e Jahn-Teller and Rabi Hamiltonian by generalised spheroidal wavefunctions? J. Phys. A: Math. Gen. 1987, 20, 6327–6340. (9) Szopa, M.; Mys, G.; Ceulemans, A. The canonical form of the Rabi Hamiltonian. J. Math. Phys. 1996, 37, 5402–5411. (10) Reik, H. G. In in: The dynamical Jahn-Teller effect in localized systems; Perlin, Y. E., Wagner, M., Eds.; Elsevier: Amsterdam, 1987; Chapter 4. (11) Bargmann, V. On a Hilbert space of analytic functions and an associated integral transform. Communications on Pure and Applied Mathematics 1961, XIV, 187–214. (12) Birkhoff, G. On a simple type of irregular singular point. Trans. Am. Math. Soc. 1913, 14, 462–476. (13) Bolibruch, A. A. On the Birkhoff Standard Form of Linear Systems of ODE. Am. Math. Soc. Transl. (2) 1996, 174, 169–176. 26 ACS Paragon Plus Environment

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(14) Turrittin, H. L. Reduction of ordinary differential equations to the Birkhoff canonical form. Trans. Am. Math. Soc. 1963, 107, 485–507. (15) Slater, L. J. Confluent hypergeometric functions; Cambridge University Press: Cambridge, 1960. (16) John W. Eaton, S. H., David Bateman; Wehbring, R. GNU Octave version 3.8.1 manual: a high-level interactive language for numerical computations; CreateSpace Independent Publishing Platform, 2014; ISBN 1441413006. (17) Moffitt, W.; Thorson, W. Vibronic States of Octahedral Complexes. Phys. Rev. 1957, 108, 1251. (18) Judd, B. R. Jahn-Teller degeneracies of Thorson and Moffitt. J. Chem. Phys. 1977, 67, 1174–1179.


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