Potential energy of neutral clusters

Neumark and coworkers^[3]included three contributions to the potential energies of the neutral clusters: pairwise interactions among the noble gas atoms, “matrix-additive”

Chapter 4. Interactions in weakly bound noble gas–halide clusters

X

Ng_i

Ng_j rⁱ

r_j

rij

θx

θi

θj

Figure 4.2:Lengths and angles for the triple dipole potential.

interactions between the halide atom and the noble gas atoms, and Axilrod-Teller triple-dipole interactions. Three-body exchange interactions were disregarded as their treatment proved too difficult and their contribution too small.^[3]

Here, this treatment was followed in general, and the potentials and parameters used are listed in Table C.1, C.3, and C.2 for the interaction of the halide atom with the noble gas atoms and Table C.4 for the pairwise interaction of the noble gas atoms among themselves, respectively. For the Axilrod-Teller triple dipole term, Yourshaw’s treatment rests on the definition of the potentialVddd, the C9constant and the atomic average excitation energyηkin the following form:

V_ddd =C₉3 cosθx cosθ_i cosθ_j r_ij³r³_i r³_j

C9= ³

2αxα_iα_j ηxηiηj(ηx+ηi+ηj) (ηx+η_i)(ηx+η_j)(η_i+η_j)

η_k = s

N_k α_k

where αk is the dipole polarizability and Nk the effective number of electrons for the given particle, respectively.^[3] All dimensions required are illustrated in Figure 4.2.

Values ofC₉were computed from the dipole polarizabilities and effective numbers of electrons of the individual particles as given in Table C.6.

In their treatment of spin-orbit coupling in clusters of electrically neutral, i.e.

open shell, halide atoms, Yourshaw et al.^[3] largely relied on the model proposed by Lawrence and Apkarian^[164]. The Hamiltonian of this model is restricted to additive

4.1. Model of the potential energy noble gas–noble gas interactions, anisotropic noble gas–halide interactions and spin-orbit coupling. The latter is assumed not to be influenced by the noble gas shell and the interaction between the halide and its noble gas shell is assumed not to impact on the interaction of the noble gas atoms among themselves.

In light nuclei, such as oxygen^[169], spin-orbit coupling may of course be neglected and the problem is thus reduced to treating the splitting of angular momentum. Par-ticularly with the heavier halides, however, spin-orbit interaction needs to be incorpo-rated. It is convenient for this purpose to express the halide–noble gas Hamiltonian in the basis of the|j m_ji-eigenstates of the open shell atom. In this representation, the spin-orbit Hamiltonian is simply a diagonal matrix that may be added to the halide–

noble gas interaction Hamiltonian. Diagonalization of this combined Hamiltonian, as given by Lawrence and Apkarian^[164], yields the eigenstates of the halide–noble gas interaction.

It is obvious, however, that an analytical solution of this problem is desirable in or-der to accelerate model calculations of the aforementioned systems. Yourshaw already commented on the considerable improvement in speed achieved thereby^[3], but the so-lution obtained at the time still proved rather unwieldy^[170]. For this work, Lawrence and Apkarian’s Hamiltonian^[164]was also used and a much simpler analytical solution to the problem of an atom with a single electron or hole in a P-state surrounded by closed-shell (S-state) atoms was derived.

Here, the following notation for the treatment of the anisotropic component of the interaction will be used

Vxy = ³ 5

∑

i

xiyi

r²_i V2 V_y²_−x² = ³ 10

∑

i

y²_i −x²_i r²_i V2

Vxz = ³ 5

∑

i

x_iz_i

r²_i V2 V_z² = ¹ 10

∑

i

3z²_i r²_i −₁

! V2

Vyz = ³ 5

∑

i

y_iz_i r²_i V₂

The respective sums run over all atoms and the isotropic part isV₀ = (V_Σ+2V_Π)/3, while the anisotropic part of the potential is V₂ = 5(V_Σ −V_Π)/3, as usually de-fined^{[3, 171]}. The degeneracy of the²P state of the halide atom being lifted in the pres-ence of a closed-shell noble gas atom, V_Σ and V_Π are the potentials of the two eigen-states of the diatomic case.^{[171, 172]}

With spin-orbit splitting included, the ²P state is itself already non-degenerate,

Chapter 4. Interactions in weakly bound noble gas–halide clusters

yielding two terms,²P1/2 and²P3/2. While the former is of pureV_Π character, the latter bears both V_Σ and V_Π contributions, and it is split by the presence of a closed-shell noble gas atom, such that the states depicted in Figure 4.1 arise. In terms of these spectroscopic potentials, the isotropic component is then V0 = ¹₃V_X1

2 +V_I³

The elements of the corresponding Hamiltonian in the|l mli-basis can then be writ-ten as:[164, 173, 174]

From here on, the isotropic componentV0 will not be considered as it is simply addi-tive. Using the Maple 12.01 analytical software^[175] it was possible to obtain analytical solutions of the eigenvalues. As in Yourshaw’s version^[170], these proved to be rather bulky. Closer inspection, however, allowed for some significant simplifications to fi-nally arrive at rather concise expressions.

The structures of the three doubly degenerate eigenvalues thus obtained were E1,2 =−¹

is obviously a positive, real number. Since the Hamiltonian is a Hermitian matrix, its eigenvalues are necessarily real. WhileE3 = a+b/amust already be real by itself, since it constitutes an eigenvalue of the Hamiltonian,a−b/amust be purely imaginary in order to render the second summand in E1,2 real and thus make E1,2 itself real as well. Combining these conditions leads to b/a = a^∗ and thus b = |a|². Introducing ar =Re(a)anda_i =Im(a), it then follows immediately that

E1,2 =−_ar∓√

3ai E3 =_{2ar (4.2)}

4.1. Model of the potential energy At this point it proved advantageous to switch to polar representation, for which the modulus of ais readily available as |a| = √

b. The associated polar angle φa then remains to be found. Closer inspection of the computer-generatedarevealed that it is of the structure

ar+i a_i ≡ |a| exp(iφa) = ^q3 cr+√

. . . = ³ q

|c| exp(iφc/3) (4.3) where

cr = ∆

3 3

+V_z² V_xz² +V_yz²

2 −V_xy² −V_y²2−x²+V_z²2

!

+VxyVxzVyz+^Vy2−x2 2

V_yz² −V_xz²

(4.4)

is also a real, but not necessarily positive number. The term abbreviated as√

. . . is the most cumbersome part ofa. While the radicand is real, it is extremely tedious to ver-ify that it is also negative.For all situations of interest, however, the opposite case of a positive radicand can be ruled out since it would lead to a realaand thus degenerate eigenvalues E2,3. It should be mentioned, however, that in some cases whereφa → 0 when the first shell of noble gas atoms around the halide atom is complete or nearly complete, one of the states could not be converged to a minimum energy geometry.

This is probably due to near-degeneracy, and these states cannot be disentangled eas-ily; using the solution from explicitly diagonalizing the Hamiltonian led to the same problem. For these cases, it would be desirable to obtain a non-periodic (see below) set of solutions.

Provided that√

. . . is imaginary, the angle ofain polar representation,φacan easily be obtained from eq. Equation 4.3:

φa = ¹

3arccos cr

|a|³ = ¹

3arccos cr

b³² (4.5)

Using some basic trigonometric operations, the eigenvalues in eq. Equation 4.2 can be cast into the form

En =₂√ b cos

φa−^4nπ 3

(4.6)

It may be worth mentioning that despite the constraint of 0 6 φa < π/3 in the determination of the polar angle via eq. Equation 4.5, the periodicity of the

eigenval-Chapter 4. Interactions in weakly bound noble gas–halide clusters

ues in eq. Equation 4.6 warrants for covering the entire eigenvalue spectrum. Thus, Equations 4.1, 4.4, 4.5 and 4.6 as working equations establish a very concise scheme for treating the interaction of a singleP-state atom with a number of closed shell atoms.

Im Dokument S2 State Photodissociation of Diphenylcyclopropenone, Vibrational Energy Transfer along Aliphatic Chains, and Energy Calculations of Noble Gas-Halide Clusters (Seite 127-132)