Eur. Phys. J. D (2013) 67: 253 DOI: 10.1140/epjd/e2013-40371-3
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Alternative hyperspherical adiabatic decoupling scheme for tetratomic molecules: quantum two-dimensional study of the ammonia umbrella motion Marcilio N. Guimar˜ aes1 , Mirco Ragni2 , Ana Carla P. Bitencourt2 , and Frederico V. Prudente3,a 1 2 3
Centro de Forma¸ca ˜o de Professores, Universidade Federal do Recˆ oncavo da Bahia, CEP: 45.300-000, Amargosa, BA, Brazil Departamento de F´ısica, Universidade Estadual de Feira de Santana, CEP: 44.036-900, Feira de Santana, BA, Brazil Instituto de F´ısica, Universidade Federal da Bahia, CEP: 40.170-115, Salvador, BA, Brazil Received 22 June 2013 / Received in final form 25 September 2013 c EDP Sciences, Societ` Published online 10 December 2013 – a Italiana di Fisica, Springer-Verlag 2013 Abstract. The prototypical example of the inversion of ammonia is studied as an approach to hyperspheric decoupling schemes for tetratomic molecule. We perform two-dimensional calculations including totally symmetric stretching and inversion modes, associated to the hyperradius and an hyperangle. Two adiabatic procedures are employed: the hyperradial one where the hyperangular part is solved to give hyperradial adiabatic curves, and the hyperangular one where the hyperradial part is solved to give hyperangular adiabatic curves. Two methodologies are employed: the hyperquantization algorithm and the finite element method. For completeness a diabatic procedure is also formulated.
1 Introduction Ammonia is a prototype pyramidal molecule which provides a good test case for theoretical spectroscopic models. In this paper we employ theoretical and computational methods to study the ammonia inversion. To describe the low-frequency large amplitude inversion motion we use hyperspherical coordinates defined from the Radau-Smith orthogonal local coordinates [1]. Here the hyperspherical approach is used in two-dimensional calculations including the totally symmetric stretching and the inversion mode, associated to the hyperradial and hyperangular coordinates, respectively. It can be shown that the hyperradius and the hyperangle are almost separable by treating the hyperradius as an adiabatic variable analogue to internuclear distances in the adiabatic (Born-Oppenheimer) approximation for molecules. Generally, in reactive problems using hyperspherical coordinates, the hyperangular part is solved initially for fixed values of the hyperradius, yielding a set of coupled hyperradial potential curves. Then, the coupled hyperradial equations are solved [2]. For the present bound state problem we have tested this procedure that here we call as usual scheme. On the other hand, we also consider an alternative adiabatic procedure. Instead of solving firstly the hyperangular part, we can solve the hyperradial problem for fixed values of the inversion angle of the molecule, yielding a set of coupled hyperangular potential curves. Then, the resultant a
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coupled hyperangular equations are solved. This alternative procedure introduces adiabatic hyperradial corrections to the umbrella potential energy curve. For solving the hyperangular part we apply the hyperquantization algorithm [3] while for the hyperradial one we employ the finite element method [4,5]. The hyperquantization algorithm had been successfully applied to study ion-molecule [6] and atom-molecule [7–9] reactions, and, more recently, to treat bound state problems [1,10]. The extremely sparse structure in the construction of the kinetic energy matrix and the fact that the potential energy matrix is diagonal is an advantage for the diagonalization required to evaluate the adiabatic hyperspherical states. On the other hand, the finite element method has been widely used in the analysis of engineering problems, but over the years has been applied in the study of molecular dynamics and electronic structure related to both bound quantum systems [5,11,12] and in quantum scattering processes [2,13]. An advantage of this method is to provide a way to systematically improve the accuracy of the calculations in a natural manner (see, e.g., Refs. [11,14]). This is the first time that both methods are used together, exploring the usefulness of the hyperquantization algorithm for solving typical differential equations encountered in the quantum chemical problems and the easy adaptability of the finite element method to solve the various types of partial differential equations found in physics, engineering, mathematics, etc. In Section 2 we present the theory where we discuss the usual and alternative adiabatic hyperspherical procedures.
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subsection (Sect. 2.1) to calculate the vibrational levels of ammonia in C3v symmetry. The hyperangular part is solved for fixed values of the hyperradius, yielding the adiabatic curves as a function of ρ. Then the low lying vibrational energy levels of ammonia are obtained from a set of hyperradial equations. However, the vibrational energies associated to the stretching mode are larger than the umbrella motion ones [15]. This suggests an alternative procedure where first it is solved the part with the largest energy, i.e., the hyperradial problem for fixed values of the hyperangle. This procedure is presented in Section 2.2. 2.1 Hyperradial adiabatic procedure As mentioned previously, generally the first step to compute the time-independent Schr¨ odinger equation with Hamiltonian given in equation (1) consists in solving the problem for fixed values of ρ. So the reduced problem is: 2 − Δ(θ) + V (θ; ρ) ψl (θ; ρ) = l (ρ) ψl (θ; ρ), (3) 2mρ2
Fig. 1. Scheme of coordinates.
In Section 3 we present and discuss the numerical methodologies. In Section 4 we present the results and compare them with ones previously available in the literature. At last, in Section 5, we present our conclusions and perspectives. Also, in the Appendix, a diabatic procedure is proposed to treat this kind of problems.
2 Theory The reduced Hamiltonian for the study of the umbrella inversion motion in C3v symmetry of the ammonia in hyperspherical coordinate is 1 2 −8 ∂ 8 ∂ ˆ ρ ρ + Δ(θ) + V (ρ, θ), H(ρ, θ) = − 2m ∂ρ ∂ρ ρ2 Δ(θ) =
∂ 1 ∂ sin θ . sin θ ∂θ ∂θ
(1)
See [1] for details. In Figure 1 the scheme of coordinates is illustrated. Basically, the hyperangle θ is the angle between the z axis and the vector that connect the canonical point E with one of the three hydrogens. The hyperradius ρ is related to the length rEH by the following equation: 3 mH rEH , (2) ρ= m where mH is the mass of hydrogen nucleus and m is the total mass of the system. As discussed in reference [1], the potential energy surface for the symmetric stretching and the inversion motion are nearly separable if represented by an appropriated function of the hyperradius ρ and the hyperangle θ. Inspired by the usual method employed in reactive problems, we discuss an analogue procedure in the next
where the eigenvalues l (ρ) represent the lth hyperradial adiabatic energy curve. The orthonormalization condition of the eigenfunctions is expressed by: −1 − ψl (θ; ρ)ψl (θ; ρ)d cos(θ) = δl,l . (4) 1
The eigenfunctions of the two-dimensional problem are searched in the form Ψn (ρ, θ) = ρ−4 hnl (ρ) ψl (θ; ρ), (5) l
where hnl (ρ) is the coefficient of the expansion. The orthonormalization of the Ψn is expressed by: −1 ∞ ρ8 dρ Ψn∗ (ρ, θ) Ψn (ρ, θ)d cos(θ) = δn,n . (6) − 0
1
The functional to be solved is then given by: −1 ∞ 8 ˆ θ) − E ρ dρ Ψn∗ (ρ, θ) H(ρ, J =− 0
1
× Ψn (ρ, θ)d cos(θ) ∞ ∂2 2 h∗l (ρ) 2 hl (ρ)dρ =− 2m 0 ∂ρ 2
− 2m
l
∞
0
l
l
l
h∗l (ρ)
∂ (1) (2) + Al,l (ρ) 2Al,l (ρ) ∂ρ
× hl (ρ)dρ
∞ 62 ∗ hl (ρ) l (ρ) + − E hl (ρ)dρ, (7) + mρ2 0 l
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where
(1)
Al,l (ρ) = −
In the present case, the functional to be solved is given by:
−1
1
ψl (θ; ρ)
∂ ψl (θ; ρ)d cos(θ), ∂ρ
(8)
ψl (θ; ρ)
∂2 ψl (θ; ρ)d cos(θ) ∂ρ2
(9)
and
(2)
Al,l (ρ) = −
1
−1
0
−
are the non-adiabatic derivative couplings. The first and a good approximation, similar to the Born-Oppenheimer approximation, consists in neglecting (i) the coefficients Al,l (ρ) and solving the remaining part of the functional. Making these considerations in the previous functional we can rewrite the problem as follows 2 d2 ef − + V (ρ) hnl (ρ) = Eln hnl (ρ), l 2m dρ2
(10)
with Vlef (ρ) = l (ρ) +
62 . mρ2
(11)
The solution of equation (10) provides the energy spectrum associated to the stretching and inversion modes. Thus, the ground state of each hyperradial potential curve Vlef will correspond to low energy states of the umbrella motion.
2.2 Hyperangular adiabatic procedure The alternative approach to solve the two-dimensional problem (1) consists in finding firstly the solutions for the reduced problem with fixed values of θ:
2 −8 ∂ 8 ∂ hj (ρ; θ) hj (ρ; θ) ρ ρ + V (ρ, θ) = j (θ) . 2m ∂ρ ∂ρ ρ4 ρ4 (12) In this case the eigenvalues j (θ) depend parametrically on θ and represent the jth hyperangular adiabatic curve. For j = 0 we have the inversion potential energy curve with adiabatic correction, namely ground hyperangular adiabatic curve. The eigenfunctions are orthonormalized: −
∞ 0
h∗j (ρ; θ)hj (ρ; θ)dρ = δj,j .
(13)
Similarly to the strategy described in the previous subsection, the wavefunctions are searched in the form Ψn (ρ, θ) = ρ−4
gjn (θ) hj (ρ; θ),
j
where gjn (θ) is the coefficient of the expansion.
−1 ∞ ˆ − E Ψn (ρ, θ)d cos(θ) ρ8 dρ Ψn∗ (ρ, θ) H J =− 1 π0 = sin θdθ (j (θ) − E) gj (θ) gj (θ)
(14)
2
2m ∞
j
j
π
sin θdθ
0
j
gj (θ) [Δ(θ) gj (θ)]
j
1 h∗j (ρ; θ) 2 hj (ρ; θ)dρ ρ 0 π 2 − sin θdθ gj (θ) gj (θ) 2m 0 j j ∞ Δ(θ) × h∗j (ρ; θ) 2 hj (ρ; θ)dρ ρ 0 2 π ∂gj (θ) − sin θdθ gj (θ) m 0 ∂θ j j ∞ ∂hj (ρ; θ) dρ. h∗j (ρ; θ) × ∂θ 0 ×
(15)
As in the Born-Oppenheimer separation, a good approximation consists in neglecting the coupling between the hyperradial adiabatic functions. This is equivalent to assume that j = j , and to solve the remaining part of the functional. This procedure leads to a better representation of the two-dimensional problem than the one described in the preceding subsection due to the larger separation in energy of the stretching mode of vibration (well approximated by ρ) with respect to the umbrella inversion one (well approximated by θ). Making these considerations in the functional (15), we can rewrite our problem as follows
2 j (θ) Ejn j − Δ(θ) + gnj (θ) = g (θ), 2m fj (θ) fj (θ) n
(16)
where fj (θ) =
0
=
0
∞
∞
1 hj (ρ; θ)dρ ρ2 |hj (ρ; θ)|2 dρ. ρ2
h∗j (ρ; θ)
(17)
Another more drastic approximation consists in fixing the hyperradius to its equilibrium value in equation (17). Thus, due to the orthonormality of the hj (ρ; θ) functions, equation (17) becomes fj =
1 . ρ2eq
(18)
The solution of equation (16) for the ground hyperangular adiabatic curve, 0 (θ), provides directly the low energy states of umbrella motion with adiabatic hyperradial corrections.
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Eur. Phys. J. D (2013) 67: 253
3 Computational methods Hyperradial and hyperangular parts of the problem are significantly different. For example, the hyperangular part gives always discrete eigenvalues, due to the limits of the hyperangles and to the boundary conditions. The solution of the hyperradial part, depending on the shape of the potential, can lead to bound state but always has a region where the energies are continuous. For this and for other reasons it is natural to use different approaches to tackle the two parts. In this work we use the finite element method to solve the hyperradial part, while the hyperquantization algorithm is used to found the solutions of the hyperangular one. These two methods are briefly reviewed in the following two subsections.
The solutions of the kinetic part of this equation are directly the Legendre polynomials Pl (θ). The first step of the HQA is to discretize them introducing the variable ξ: − cos(θ) =
PlM (ξ) = K(l)
(19)
where fji (ρ) is the jth basis function of the ith element. In particular, we employ the p-version of FEM (p-FEM) [2,4,5,11,13,14] which utilizes as basis functions two linear interpolating functions defined by ρi − ρ ρi − ρi−1 ρ − ρi i fki (ρ) = , ρi − ρi−1
M
PlM (ξ)PlM (ξ ) = δξ −ξ .
(26)
The matrix elements are calculated inverting the roles of the discrete variable, ξ, and the degree of the Legendre Polynomials, l. Consequently the matrix elements are calculated as: M 2 M H ξ ,ξ = PlM (ξ ) − Δ (ξ) + V (ξ) PlM (ξ) 2mρ2 l=0 2 M = PlM (ξ ) l(l + 1) + V (ξ) PlM (ξ) 2mρ2 l=0
=
M 2 l(l + 1)PlM (ξ )PlM (ξ) + V (ξ)δξ ,ξ . 2 2mρ l=0
(27) (20)
It is known from the literature [16] that the recurrence equation a(ξ)PlM (ξ − 1) + b(ξ) − f M (ξ) l(l + 1) PlM (ξ) + c(ξ)PlM (ξ + 1) 0 (28)
(21)
where j = 1, . . . , ki − 1, xi = 2f0i (ρ) − 1 and Pj are the Legendre polynomials. Using the property (19), the ma has the following trix representation B of any operator B feature ρNe ii
fji (ρ) = 0, ∀i = i . Bjj = dρfji (ρ)B (22)
is true for M → ∞ and M M2 − , 4 2 2 M + M, b(ξ) = −2ξ 2 + 2 M M2 − . c(ξ) = ξ 2 + ξ − 4 2
a(ξ) = ξ 2 − ξ −
ρ0
Note that the resulting B matrix is very sparse, concentrated on the diagonal and with a block structure.
The differential problem soluble by hyperquantization algorithm (HQA) has the form: =
2
(23)
(29)
This recurrence relationship permits to write matrix elements in a simple form: H ξ ,ξ =
3.2 Hyperquantization algorithm
ˆ H(θ) =− Δ(θ) + V (θ). 2mρ2
(25)
For M → ∞ we have
f0i (ρ) =
and ki − 1 shape functions defined by 1 fji (ρ) = (4j + 2)− 2 Pj+1 (xi ) − Pj−1 (xi ) ,
l dl 2 x −1 . l dx
l=0
The finite element method (FEM) in the one-dimensional case consists in dividing the integration interval [ρ0 , ρNe ] into Ne elements, being the ith element defined in the range of ρi−1 to ρi , and the wave function is expanded in basis functions {fji (ρ)} satisfying the following property if ρ ∈ / [ρi−1 , ρi ],
(24)
where ξ = −M/2, −M/2 + 1, . . . , M/2 − 1, M/2 and M is the number of intervals of the grid. Being K(l) the normalization factor, then
3.1 Finite element method
fji (ρ) = 0
2ξ = x, M
∞
2 M Pl (ξ ) a(ξ)PlM (ξ − 1) + b(ξ)PlM (ξ) 2m l +c(ξ)PlM (ξ + 1) + V (ξ)δξ ,ξ , 2 [a(ξ)δξ ,ξ−1 + b(ξ)δξ ,ξ + c(ξ)δξ ,ξ+1 ] 2m + V (ξ)δξ,ξ .
(30)
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The matrix to be diagonalized is tridiagonal as the one built by Lanczos method, so fast computational algorithms can be implemented [17]. Note that the procedure to solve equation (3) is well established in literature, but the one to solve equation (16) is not. Thus, we give here how to build up the matrix representation for the case described in Section 2.2 given by equation (16), indicated by H and obtained expanding the eigenfunction with an appropriate basis set of functions. First we note that H is not symmetric, and its matrix elements are given by equation (30) with V (ξ) =
j (ξ) . fj (ξ)
(31)
To solve this problem the eigenvalues are multiplied by a diagonal matrix S with diagonal elements given by S
ξ ,ξ
= (S
ξ ,ξ 1/2
)
(S
ξ ,ξ 1/2
)
1 δξ,ξ . = fj (ξ)
(32)
Fig. 2. Hyperradial adiabatic curves as a function of rEH . The valley bottom: blue line. Ridge: red line. Black lines are the adiabatic curves, in particular the dashed (dotted) ones are given by the symmetric (anti-symmetric) solutions.
Now observe that the problem can be symmetrized rearranging it as follows. H C = E S C, −1/2
S
−1/2
HS
1/2
S
C = E S−1/2 S C,
S−1/2 H S−1/2 C = E C , H C = E C ,
(33)
where we indicate with C the eigenvectors. Finally the H matrix of equation (33) is related with H ones (Eq. (30)) as (34) H ξ ,ξ = f (ξ ) f (ξ) H ξ ,ξ . The matrix is also centro-symmetric and is factorized in two sub-matrices (symmetric and anti-symmetric solutions). Fig. 3. Zoom of Figure 2.
4 Results In this section we apply the theoretical and computational methods developed over the previous sections to study the umbrella inversion motion of the NH3 molecule. We perform three calculations using the two-dimensional potential energy surface from Pesonen et al. [18] calculated with the CCSD(T)/aug-cc-pVQZ level of theory. In Figures 2 and 3 the adiabatic hyperradial curves are depicted as a function of rEH which is related to the hyperradius by equation (2). These adiabatic curves, l (ρ), was obtained by solving equation (3). The symmetric and antisymmetric eigenvalues are plotted along the valley bottoms (the minimum energy) and ridges (the corresponding barriers heights, associated to planar configuration D3h ). Figure 4 reports the adiabatic hyperangular curves as a function of θ, j (θ), obtained by solving equation (12). The ground adiabatic potential was used in equation (16) to calculate the inversion levels.
Table 1 gives the eigenvalues of the umbrella motion (fundamental stretching) calculated with the following procedures: (a) unidimensional problem (Eq. (3)) with ρ at equilibrium; (b) hyperradial adiabatic procedure, fundamental levels of equation (10) for each hyperradial potential curve Vlef ; (c) hyperangular adiabatic procedure, energy levels of equation (16) for the ground hyperangular adiabatic curve, 0 (θ), with approximation (18); and (d) similar to (c), but with fj (θ) calculated exactly from equation (17). For comparison the theoretical results of Pesonen et al. [18] obtained with the same potential energy surface are also shown in Table 1. According to notation in the literature, in Table 1 we indicate the symmetrical and anti-symmetrical levels, respectively, by lν2+ and lν2− , where l enumerates the levels. The first two levels are indicated by GS+ (symmetric ground state) and GS− (anti-symmetric ground state). In general, we can see in the Table that our two-dimensional
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Eur. Phys. J. D (2013) 67: 253 Table 2. Umbrella inversion energy levels for first excited stretching mode. Values in cm−1 . GS+ GS− ν2+ ν2− 2ν2+ 2ν2− 3ν2+ 3ν2− 4ν2+ 4ν2−
Fig. 4. Hyperangular adiabatic curves. Umbrella inversion path predicted by ρ of equilibrium: blue line. Optimized: dots. Red lines: adiabatic curves calculated solving the hyperradial equation with the p-FEM for each value of θ. Table 1. Umbrella inversion energy levels for fundamental stretching mode. Values in cm−1 . GS+ GS− ν2+ ν2− 2ν2+ 2ν2− 3ν2+ 3ν2− 4ν2+ 4ν2−
(a) 0.00 1.26 904.50 956.53 1545.69 1877.79 2389.00 2925.40 3512.79 4137.07
(b) 0.00 1.27 914.15 963.71 1581.65 1887.36 2402.23 2919.63 3495.73 4105.60
(c) 0.00 1.01 921.59 965.65 1574.50 1887.23 2396.49 2925.58 3507.89 4127.05
(d) 0.00 1.01 920.16 963.75 1573.01 1883.57 2391.70 2918.74 3498.95 4115.69
Ref. [18] 0.00 0.96 922.92 964.74 1577.97 1882.32 2387.96 2909.76 3485.55 4093.93
(a) Unidimensional problem solved for ρ of equilibrium (Ref. [1]). Values taken in relation to zero-point energy (ZPE) that is 506.53 cm−1 . (b) Hyperradial adiabatic procedure, ground energy level of stretching for each adiabatic curve of Figure 2. ZPE = 2306.18 cm−1 . (c) Hyperangular adiabatic procedure, energy levels of equation (16) for 0 (θ) with ρ at equilibrium value. ZPE = 2309.56 cm−1 . (d) Hyperangular adiabatic procedure, energy levels of equation (16) for 0 (θ) with fj (θ) calculated exactly. ZPE = 2308.23 cm−1 .
calculations are closer to the Pesonen et al. results than the one-dimensional calculation performed in reference [1] with the hyperquantization algorithm. In particular, the mean relative deviation of each procedure with respect the Pesonen et al results is approximately 4.3%, 4.0%, 0.93%, and 0.77% for (a), (b), (c), and (d) approaches. This indicate clearly that the alternative hyperangular procedures provide the best results for the umbrella inversion movement. The umbrella inversion energy levels for first excited stretching mode from the different two-dimensional procedures are shown in Table 2, where the experimental ones [19] are considered for comparison. In such a case, the present energy levels are calculated by using: (a) the hyperradial adiabatic procedure, first excited levels of equation (10) for each hyperradial potential curve Vlef ;
(a) 3433.66 3434.95 4364.18 4410.48 5081.78 5343.32 5880.27 6359.70 6925.52 7516.76
(b) 3432.50 3433.13 4390.37 4420.64 5079.26 5348.78 5858.51 6371.81 6945.34 7555.63
(c) 3429.54 3430.14 4381.72 4410.34 5069.41 5330.59 5835.87 6341.59 6908.01 7510.55
Exp. [19] 3336.02 3337.08 4294.51 4320.06
(a) Hyperradial adiabatic procedure, first excited level of stretching mode for each adiabatic curve of Figure 2. Values taken in relation to zero-point energy (ZPE) that is 2306.18 cm−1 . (b) Hyperangular adiabatic procedure, energy levels of equation (16) for 1 (θ) with ρ at equilibrium value. ZPE = 2309.56 cm−1 . (c) Hyperangular adiabatic procedure, energy levels of equation (16) for 1 (θ) with fj (θ) calculated exactly. ZPE = 2308.23 cm−1 .
(b) the hyperangular adiabatic procedure, energy levels of equation (16) for the first excited hyperangular adiabatic curve, 1 (θ), with approximation (18); and (c) similar to (b), but with fj (θ) calculated exactly from equation (17). We noticed a good agreement between our results and the experimental ones.
5 Conclusion and perspectives In this paper we studied the umbrella inversion motion of the ammonia molecule. We took advantage of the appealing representation of the umbrella inversion problem through the use of hyperspherical coordinates. In particular, the hyperangle θ well represents the umbrella inversion while the hyperradius ρ well describes the symmetric stretching mode. We present two different approaches to treat the two-dimensional problem that we have named as hyperradial adiabatic and hyperangular adiabatic procedures. In the second (and novel) procedure, first the hyperradial part of the problem for fixed values of the inversion angle of the molecule is solved. This is motivated by the separation in energy levels of the stretching mode is larger than in the umbrella inversion ones. The p-version of the finite element method was implemented to solve the hyperradial part and the hyperquantization algorithm with variable coefficients is developed to solve the hyperangular part. Both procedures give good results, but the hyperangular adiabatic procedure provides a better description of the physical problem. Additionally, in the Appendix we have proposed a hyperangular diabatic procedure that we have the intention to apply in the near future. With respect to our perspectives, we intend so much to perform the calculation considering the couplings between the (hyperradial or hyperangular) adiabatic functions as to implement the diabatic procedure discussed
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in the Appendix. Moreover, we can apply these methodologies to study other molecules that have the umbrella inversion motion as H3 O+ [1] and PH+ 3 [20]. Furthermore, we will seek to make a complete treatment of the NH3 and derived molecules in local orthogonal coordinates through methodologies that do not imply the numerical calculation of the integrals. For this, we expect to use the diagonalization-truncation method widely used for bound [5,21] and scattering [22,23]. These will be accompanied by the development of semi-classical expression for the partition functions [24] and the calculation of the intermolecular dynamics of ammonia interacting with noble gases. This work has been supported by the following Brazilian National Research Councils: Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico (CNPq), Coordena¸ca ˜o de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES) and Funda¸ca ˜o de Amparo a Pesquisa do Estado da Bahia (FAPESB).
where
¯ ΔV = V (ρ, θ) − V (ρ, θ).
Starting from this result the following functional is found: J =−
=
j
∞
d cos(θ) −π/2
2 2m −
0
π/2
−π/2 j,j π/2
−π/2 j,j
−
π/2
−π/2
j
ˆ −E Ψ (ρ, θ) ρ8 dρΨ ∗ (ρ, θ) H
(1)
Uj ,j gj∗ (θ) Δ(θ) gj (θ)d cos(θ) (2)
Uj ,j (θ) gj∗ (θ) gj (θ)d cos(θ) ¯ − E gj (θ)d cos(θ), (A.7) gj∗ (θ) j (θ)
where (1)
In the diabatic procedure the solutions of equation (1) are found in the form ¯ gj (θ), hj (ρ, θ) (A.1) Ψ (ρ, θ) = ρ−4
π/2
Uj ,j (θ) =
Appendix: Diabatic procedure
(A.6)
(2)
Uj ,j (θ) =
∞
0
0
∞
¯ ρ−2 hj (ρ, θ)dρ, ¯ h∗j (ρ, θ) ¯ ¯ h∗j (ρ, θ)ΔV (ρ, θ)hj (ρ, θ)dρ.
(A.8)
References
¯ h (ρ,θ)
where j ρ4 are solutions of the hyperradial operator
¯ ¯ 1 d 8 d 2 ¯ hj (ρ, θ) = j (θ) ¯ hj (ρ, θ) ρ + V (ρ, θ) − 8 4 4 2m ρ dρ dρ ρ ρ (A.2) ¯ This equation can be easily simplified in for θ fixed to θ. 2
2 12 d ¯ ¯ = j (θ)h ¯ j (ρ, θ). ¯ − − 2 + V (ρ, θ) hj (ρ, θ) 2m dρ2 ρ (A.3) ¯ are also orthonormal: The hj (ρ, θ) ∞ ¯ j (ρ, θ)dρ ¯ h∗j (ρ, θ)h = δj,j . (A.4) 0
So the differential becomes: ˆ − E Ψ (ρ, θ) = H
gj (θ) −2 d2 12 ¯ ¯ − 2 + V (ρ, θ) hj (ρ, θ) 4 2 ρ 2m dρ ρ j ¯ gj (θ) ¯ − E ρ−4 hj (ρ, θ) + V (ρ, θ) − V (ρ, θ) j
2 ¯ − hj (ρ, θ)Δ(θ)g j (θ) = 0. 2mρ6 j Simplify hj (ρ, θ) ¯ −2 ¯ Δ(θ) + ΔV + ( θ) − E gj (θ) = 0, j ρ4 2mρ2 j (A.5)
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