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Perturbation Method to Calculate the Interaction Potentials and Electronic Excitation Spectra of Atoms in He Nanodroplets Carlo Callegari*,†,^ and Francesco Ancilotto‡,§ †
Institute of Experimental Physics, TU Graz, Petersgasse 16, A-8010 Graz, Austria, EU Dipartimento di Fisica ‘G. Galilei’, Universita di Padova, via Marzolo 8, I-35131 Padova, Italy § CNR-IOM-Democritos, I-34136 Trieste, Italy ‡
ABSTRACT: A method is proposed for the calculation of potential energy curves and related electronic excitation spectra of dopant atoms captured in/on He nanodroplets and is applied to alkali metal atoms. The method requires knowledge of the droplet density distribution at equilibrium (here calculated within a bosonic-He density functional approach) and of a set of valence electron orbitals of the bare dopant atom (here calculated by numeric solution of the Schr€odinger equation in a suitably parametrized model potential). The electronhelium interaction is added as a perturbation, and potential energy curves are obtained by numeric diagonalization of the resulting Hamiltonian as a function of an effective coordinate zA (here the distance between the dopant atom and center of mass of the droplet, resulting in a pseudodiatomic potential). Excitation spectra are calculated for Na in the companion paper as the FranckCondon factors between the v = 0 vibrational state in the ground electronic state and excited states of the pseudodiatomic molecule. They agree well with available experimental data, even for highly excited states where a more traditional approach fails.
’ INTRODUCTION The electronic excitation spectra of several dopant atoms trapped in/on He nanodroplets (HeN; N ∼ 104 typically) have been recorded in recent years. The main transitions of Li,13 Na,15 K,13,6 Rb,3,68 Cs,3,6,9 Mg,1012 Ca,13,14 Ba,14,15 Sr,13,14 Ag,12,1619 Al,20 Eu,16,21 and In16 have been recorded and are invaluable to understand the exact solvation behavior of each individual dopant species, which in turn critically depends on the delicate balance between Hedopant and HeHe interaction.22 Alkali metal atoms (Li, Na, K, Rb, Cs) are of particular interest in that they alone do not solvate inside of the droplet; their equilibrium position is on the surface of the droplet. This surface undergoes a slight deformation (a dimple), which minimizes the total energy of the system (the energy cost of deforming the surface is more than compensated for by the weakly attractive alkaliHe van der Waals interaction). Incidentally, the groundstate wave function of the dopant’s valence electron is also deformed, resulting in a measurable change of the hyperfine constant.23 Our ability to explicitly calculate the distorted wave function (one of the byproducts of the calculations presented here, see the section entitled “Wave functions and Potential Energy Curves”) allowed us to correctly estimate this change.23,34 Let us note here that several atoms and molecules have also been studied in bulk solid and liquid helium; a recent review can be found in ref 25, and let us also mention newer work on alkali atoms26 and on Cu atoms and dimers.27 Of particular relevance here are detailed studies of the electronic structure of Rb and Cs atoms trapped in liquid and solid He by Kinoshita r 2011 American Chemical Society
et al.28,29 and by Hofer et al.,30,31 who use an approach similar to that presented here to calculate the excitation spectra of their target systems; another similar method was used by Eloranta32 to model dynamics of He2* excimers after optical excitation in bulk liquid He. For alkali-doped helium droplets, excitation spectra to the first excited state have previously been successfully simulated with the following approach: 1. Calculate a realistic He density F0 at equilibrium in the potential of the dopant. Either an infinite surface1 or a finite droplet3,33,34 have been used. 2. Calculate the perturbed lower and upper energy levels of the atom by convoluting the helium density with the appropriate Hedopant pair potentials (e.g., from ref 35). This is done as a function of an effective coordinate zA (the distance between dopant atom A and the center of mass of the droplet or, in the case of an infinite surface, between the dopant atom and the surface itself), resulting in a so-called pseudodiatomic potential. The helium density may be updated to its new equilibrium value for each zA (in a different context rotation of dopant molecules inside of He nanodroplets this assumption has been termed “adiabatic Special Issue: J. Peter Toennies Festschrift Received: November 22, 2010 Revised: March 7, 2011 Published: March 24, 2011 6789
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Figure 1. Coordinate systems used in this work and representative calculated density (NaHe2000, with the Na atom in its ground electronic state). The spherical coordinates (r,θ,j) are centered on the alkali metal, and the Cartesian coordinates (x,y,z) are centered on the He droplet. The coordinates {j,y} (azimuthal angle and out-ofplane direction, respectively) are not shown. The He density has been symmetrized around j for smoothness. The Na atom at its equilibrium position is shown as a gray disk. The Nasurface distance is 5.6 Å. The depth of the dimple relative to the unperturbed spherical surface of a pure droplet is 2.6 Å (the droplet surface is defined as the locus of the points where the density has decreased to 50% of the value at the center of the droplet, 0.02247 Å3).
following”36) or kept constant (“frozen droplet”, used here), corresponding, respectively, to the limit cases of a droplet adjusting infinitely fast or infinitely slow to the motion of the dopant. Note that both are approximations in that they neglect the kinetic energy of the helium; the frozen droplet is usually preferred as computationally less demanding, although it is expected to be less realistic for the heavier alkalis.1 3. Calculate the excitation spectra as the relative transition probabilities between the v = 0 vibrational state in the ground electronic state and excited states of the pseudodiatomic molecule. These probabilities are calculated at a different levels of accuracy; typically, they are approximated with the FranckCondon factors alone between the states of interest. Reference 33 uses an improved version of the Lax semiclassical method. Reference 34 uses a full threedimensional diatomics-in-molecules (DIM) approach to properly address the zero-point motion of a light dopant such as Li. Item 2 is only justified for well-isolated electronic states, that is, when the helium perturbation induces a negligible interaction between configurations. Note that mixing of configurations is explicitly included in the calculation of the pair potentials of ref 35 and thus implicitly included to some extent in the convolution. The availability of spectra of higher excited states for Na, presented in the companion paper,37 and K, Rb, and Cs6 calls for a different approach where mixing of configurations is explicitly allowed. The method that we present here is conceptually the same as that used by Pascale35 to calculate the Healkali pair potentials, namely, the system is decomposed into three parts (the valence electron, the alkali positive core, and the helium),
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whose mutual interactions are suitably parametrized, and the Schr€odinger equation of the electron is solved in the BornOppenheimer approximation for a set of fixed values of the internuclear separation. The actual solution could be sought by brute-force solving of the eigenvalue problem on a real-space grid; a two-dimensional grid is sufficient due to the cylindrical symmetry of the problem. We use instead the valence electron orbitals of the bare atom as a basis set, explicitly calculate all of the matrix elements of the Hamiltonian in this basis, and diagonalize the resulting matrix. The mixing coefficients between states are a useful byproduct of the diagonalization. We use atomic units for all calculations and Å and cm1 to display the results.38 This paper is organized as follows. The Basis Set section defines the basis set of valence electrons orbitals; the Helium Density section summarizes the bosonic-He density functional (DF) approach; these are then combined in the Perturbation Matrix Elements section to calculate the perturbation matrix elements. The energy matrix of the perturbed alkali atom is diagonalized in the Wave functions and Potential Energy Curves section; the eigenvalues and eigenvectors are then used to calculate the potential energy curves of the AHeN system and ancillary quantities. The Conclusions and Summary section comments and summarizes our results. The coordinate systems used and a representative He density are shown in Figure 1.
’ BASIS SET The valence electron orbitals for an isolated alkali metal atom A, written in a spherical coordinate system (r,θ,j) centered on the atomic nucleus, are labeled by n,l,m, the principal, orbital angular momentum, and magnetic quantum number, respectively, and have the form Ψnl(r)Ylm(θ,j), where Ylm are spherical harmonics and Ψnl is the radial part of the wave function, to be calculated numerically (we also define χnl = rΨnl). This numerical calculation has been done by several authors mostly for the sake of calculating transition probabilities, lifetimes, or other properties of alkali metal atoms.3946 Kinoshita et al.28,29 and Hofer et al.30,31 did it for the specific purpose of calculating the electronic excitation spectra of alkali atoms in bulk He. Authors differ in their choice of model potential, namely, how it is parametrized, whether or not it is l-dependent, and whether the true eigenfunctions are sought (not necessarily at the exact experimental energies) or instead a solution at the exact experimental energies (not necessarily true eigenfunctions). For all elements, except Cs, we follow the approach of Laughlin44 and reparametrize his l-dependent model potential to accurately match the experimental eigenvalues47 (we label these eigenvalues and eigenfunctions equivalently with the subscript nl or with the orbital, thus E30 E3s, E31 E3p, E40 E4s, etc.). Let us note that very sophisticated methods beyond the model potential approach have been applied to the calculation of the properties of alkali metal atoms.48,49 In the Perturbation Matrix Elements and Wave functions and Potential Energy Curves sections, we include the following orbitals, ordered by increasing energy, in the basis set: Na: {3s, 3p, 4s, 3d, 4p, 5s, 4d, 4f, 5p, 6s, 5d, 5f, 5g, 6p, 7s, 6d, 6f, 6g, 6h, 7p, 8s, 7d, 7f, 7g, 7h, 7i, 8p, 9s, 8d, 8f, 8g, 8h, 8i, 9p, 10s} K:{4s, 4p, 5s, 3d, 5p, 4d, 6s, 4f, 6p, 5d, 7s, 5f, 5g, 7p, 6d, 8s, 6f, 8p, 7d, 9s, 7f, 9p, 8d, 10s, 8f, 10p} Rb: {5s, 5p, 4d, 6s, 6p, 5d, 7s, 4f, 7p, 6d, 8s, 5f, 5g, 8p, 7d, 9s, 6f, 9p, 8d, 10s, 7f, 10p, 9d, 11s, 8f, 11p, 10d, 12s, 12p} 6790
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Table 1. Parameters Defining Vp l
Rd
λ
ϑ
0
0.946
2.73
1.0 2.05
1.44360
0.0342528 þ0.145257
1
0.946
2.73
1.0 2.05
1.62807
þ0.546041
2
0.946
2.73
1.0 2.05
0.404160 þ0.144814
>2 0.946
2.73
1.0 2.05 þ10.3055
r1
b0
b1
b2
Na 0.0457050 0.0117045
3.26797
þ0.234178
K 0
5.49
11.98
0.8 2.85
2.01822
þ0.508375
0.00664648
1
5.49
11.98
0.8 2.85
1.85354
þ0.574702
0.0456346
2 5.49 >2 5.49
11.98 11.98
0.8 2.85 0.8 2.85
1.93590 þ0.674322 0.0537747 þ0.192051 0.0624162 þ0.00467076 Rb
0
8.976 41.1844 0.7 3.75
2.60454
þ0.723395
0.0323541
1
8.976 41.1844 0.7 3.75
2.51293
þ0.911340
0.0820615
2
8.976 41.1844 0.7 3.75
2.38953
þ0.793689
0.0619553
>2 8.976 41.1844 0.7 3.75
Vp(r) has the form44
0.293058 þ0.0670698 0.00335888
Cs: {6s, 6p, 5d, 7s, 7p, 6d, 8s, 4f, 8p, 7d, 9s, 5f, 5g, 8d, 10s, 6f, 6g, 9p} Note that energy is the only selection criterion, that is, all orbitals below a maximum energy have been included. The basis set was chosen to be as large as possible, compatibly with computational time and the existence of spectroscopic data to tune the effective core potential presented in next section. The basis set is largest for Na, for which we seek to reproduce a large number of spectra in He droplets37 up to the ionization limit, and it is smallest for Cs, for which less spectroscopic data exist for the bare atom. At present, the major limitation of the model lies not on the size of the basis set but on the approximations made in the model Hamiltonian. Effective Core Potential. Na, K, Rb. The valence electron of the alkali atom is subjected to the following effective core potential V ðrÞ ¼
Figure 2. Difference between experimental47 and calculated energy levels of Na, K, and Rb atoms.
Z Zc 2ζa ðrÞ þ Vp ðrÞ r r
ð1Þ
6 8 1 Rd 1 λ ½1 eðr=r1 Þ 6 ½1 eðr=r1 Þ 4 2 r 2r þ ðb0 þ b1 r þ b2 r 2 Þeϑr ð5Þ
Vp ðrÞ ¼
and is a parametrized correction to explicitly account for core polarization effects (r1 is a scaling length, Rd is the dipole polarization of the core, and λ is an effective parameter accounting for higher-order and dynamical effects, most often treated as a free parameter). Notation varies considerably between authors for the remaining parameters (b0,b1,b2,ϑ) which are 1-dependent, but the functional form used is essentially the same and merely represents a smooth exponentially decaying correction to get the calculated eigenvalues to better agree with the experimental values. Our optimized parameters defining Vp for each of the alkali atoms are reported in Table 1; Figure 2 shows the difference between experimental and calculated eigenvalues. Cs. No tabulated core HartreeFock orbitals are available for Cs. Instead, we use, without further modification, the model potential of Schweizer,46 where V(r) has the form
where Z is the nuclear charge and Zc the number of core electrons (Z Zc = 1 here); ζa(r) accounts for the core field and is calculated as follows from core HartreeFock orbitals ð2ζlj Þnlj þ 1=2 Sðζlj , nlj , rÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi r nlj 1 eζlj r ð2nlj Þ! Ψcnl ðrÞ ¼ Z
¥
ζa ðrÞ ¼ r
∑j Cnlj Sðζlj, nlj, rÞ
dx ðx2 rxÞ
ð2Þ ð3Þ
ð2l þ 1ÞΨcnl ðxÞ2 ∑ n, l 50
ð4Þ
where Cnlj, ζlj, and nlj are those tabulated by Koga and x is a dummy variable. A key to such standardized tables can be found in ref 51; note the distinction between n, which is the principal quantum number of the core orbital Ψcnl(r), and nlj, which are tabulated integer coefficients defining the Slater-type orbitals S(ζlj,nlj,r). The sum in eq 4 extends over all core orbitals Ψcnl; note that we have explicitly indicated the (2l þ 1) multiplicity, so that 2ζa(0) equals the number of core electrons.
1 V ðrÞ ¼ ½1 þ ðZ 1Þea1 r þ a2 rea3 r r
ð6Þ
with a1 = 3.294, a2 = 11.005, and a3 = 1.509. In this case, we tolerate some deviation between the calculated and experimental eigenvalues, which, on average (n up to 40), is 0.60.7%46 but is as high as 3.6% for the 6s ground state; in comparison, Hofer et al. find for their model30 that levels up to n = 12 agree within 0.5% with their experimental values. We obtained from the authors of ref 30 the wave functions χ6s and χ6p (the latter without inclusion of spinorbit), which, away from the nodes, differ by about 1% from ours; in particular, their height of the outer maximum differs by þ1.6% for χ6s and by þ0.08% for χ6p. Their position of the outer maximum differs by 0.8% for χ6s and by þ3% for χ6p. Finally, their value of |Ψ6s(0)|2 is larger by 10%; Hofer et al. use this quantity to calculate the hyperfine splitting for the free Cs atom (including unspecified correction factors to account for relativistic effects and electrostatic and magnetic volume corrections) and obtain a value ≈6% larger than the experimental one. Because the leading contribution to the hyperfine splitting is proportional to |Ψ6s(0)|2, we presume that ours would be smaller by about 4% than the experimental value. 6791
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Figure 3. Calculated wave functions χ(r) for the s states of Na, from 3s (black, innermost) to 10s (blue, outermost).
An in-depth analysis of the accuracy of these wave functions is beyond the scope of this work; we just want to point out that it is sufficient to assign the experimental spectra observed so far.6 Because the main target of the present model is higher excited states, we want to observe that, in fact, the accuracy of the unperturbed one-electron wave functions naturally increases as they more closely resemble those of the hydrogen atom, whereas the effects of approximations made in the calculation of the interaction of the alkali atom with the helium become more severe. In conclusion, while a standardized treatment of the unperturbed wave functions is desirable, an evolution of the present model should, in our opinion, concentrate first on improving the model Hamiltonian, as discussed in the sections Perturbation Matrix Elements and Wave functions and Potential Energy Curves. For reference, let us note that in the matrix E 0 of the Wavefunctions and Potential Energy Curves section, we use the experimental eigenvalues. Numerical Solution. The numerical solution of the Schr€ odinger equation for Ψnl is trivial nowadays even for the nonspecialist, with software packages such as Mathematica that we use here in combination with the Cooley algorithm.52 We solve the equation for χ lðl þ 1Þ 00 χ ðrÞ þ 2 Enl V ðrÞ χðrÞ ¼ 0 ð7Þ 2r 2 with the sign convention that χ > 0 at large r. We integrate outward from r = 103a0, inward from r = 300 a0, and splice the solutions at the half-height point inward of the outermost maximum [if rOM,nl is the position of the outermost maximum of χnl and rS,nl < rOM,nl that of the splice point, then χnl(rS,nl) = χnl(rOM,nl)/2]. For Cs only, the inner integration limit is 105a0 (except for 0.5a0 for f states and 102a0 for g states), and the outer one varies between 50 and 200a0. As a test that the calculated wave functions (those corresponding to the s state of Na are shown in Figure 3) are reasonable, we plot their behavior in terms of well-known scaling laws for Rydberg atoms (see, e.g., refs 53 and 54). Accordingly, the binding energy Enl > 0 of a Rydberg electron, in atomic units, defines an effective quantum number n*l = 1/(2Enl)1/2 that, in general, is noninteger and l-dependent, and the value of most observables scales as a known power of nl*. Specifically, we calculate the value of |Ψ(0)|2 and that of Æræ for the s states (Figure 4) and plot them
Figure 4. Calculated values, in atomic units, of Æræ and |Ψ(0)|2 for s states of Na (black), K (blue), Rb (green), and Cs (red). The solid lines are best fits to (En0)1 and (En0)3/2, respectively.
against En0. The first observable is, by definition, most sensitive to 2 the short-range part of the potential and scales as (n*) l ; thus, 1 (Enl) . The second observable is most sensitive to the longrange part of the potential and scales as (nl*)3; thus, (Enl)3/2.
’ HELIUM DENSITY We calculate the minimum-energy configuration of a droplet consisting of N = 2000 helium atoms subjected to the external potential of an alkali atom A in the electronic ground state. The latter is taken to be the AHe pair potential calculated by Patil55 and indicated here with VAHe. For convenience of notation, we move away from the spherical coordinates system centered on the alkali atom of the Basis Set section. We define a system of Cartesian coordinates (x,y,z) with origin at the center of mass of the droplet and the alkali atoms at (0,0,zA). We use here a DF approach developed for bosonic-He and widely employed in recent years to study 4He in confined geometries (for a recent review of applications to helium cluster physics, see ref 56), and which is based on the energy DF originally proposed for liquid 4He by Dalfovo.57 This DF approach represents a valid alternative to a full quantum description of inhomogeneous (and often spatially extended) 4He systems, which might be computationally very demanding. In our calculations, performed at zero temperature, we used a supercell geometry, with periodic boundary conditions on the 4 He wave function ΨHe(x,y,z) [F(x,y,z)]1/2 and on the 4He density F(x,y,z) itself. Both ΨHe and F are expanded in plane waves, whose maximum number is chosen such as to give converged values for the total energy and for the structural properties of the He droplets. This allows the use of fast Fourier transform techniques to efficiently compute the convolution integrals entering the definition of the energy DF. The selfconsistent iterative solution of the EulerLagrange equation 6792
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resulting from the minimization of the energy DF provides the extremal density profile F0(x,y,z) of the droplet. The computational method used for the functional minimization of the total energy DF is described in ref 58, to which the reader is referred for further details. The final output is a helium density F0(x,y,z) calculated on a uniformly spaced Cartesian grid, with spacing (100/192)Å ≈ 0.52 Å (Figure 1). Although all integrals in the next section are calculated on this grid, cylindrical symmetry is exploited to determine those that are strictly zero. Specifically, only matrix elements between orbitals with the same magnetic quantum number m are computed. Because we are interested in the use of these AHeN potentials to simulate one- and two-photon excitation spectra from the ground electronic state, we direct our computational effort to m = 0, 1, and 2.
’ PERTURBATION MATRIX ELEMENTS The reference system is the same as that in the Helium Density section. Let us remark the fact that the density will be “frozen”, that is, not reoptimized, when the alkali atom is moved from its equilibrium position; its dependence on x,y,z will be implicit from now on, that is, F0 F0(x,y,z). The alkalidroplet groundstate interaction potential V0(zA) as a function of the alkali position zA is calculated as Z V0 ðzA Þ ¼ F0 VAHe ðr 0 Þ dx dy dz ð8Þ with VAHe as the pair potential introduced in the Helium Density section and r0 = [x2 þ y2 þ (z zA)2]1/2. By definition, V0 = 0 for zA f ¥. As we shall see, we will use V0(zA) as a reference; that is, beyond those implicitly contained in eq 9, we will not explicitly calculate any polarization interactions (notably, the interaction between the positive alkali core and the He droplet). We are aware that this results in the wrong limit behavior for very highly excited states, where the main correction is a decrease of the total energy due to polarization of the He by the positive alkali core (experimentally observed for Na atoms as a lowering of the ionization energy59). One could include polarization effects by brute-force calculation of three-body eHecore polarization interactions, at substantial computational cost; we are instead testing improved semiempirical corrections. The dropletelectron perturbation matrix M is calculated with the local part of the eHe interaction potential of Cheng et al.60 Because the largest contribution to the perturbation should originate from the inside of the droplet, where the density is uniform, we expect that skipping the nonlocal part should not be critical, whereas it considerably speeds up the computation; a posteriori, the reasonable agreement with the experimental spectra supports our decision. The eHe interaction is written as a function of the local helium density60 1=3 p2 k20 2πp2 4π VeHe ðFÞ ¼ þ FaR 2πRe2 F4=3 ð9Þ 3 2me me where me is the electron mass, R = 0.208a30 is the static polarizability of a 4He atom, and k0 as a function of F is determined from the helium local WignerSeitz radius rs = (3/4πF)1/3 by solving the eigenvalue equation tan[k0(rs ac)] = k0rs, with ac and aR as the scattering lengths arising, respectively, from a hard-core and a polarization potential. We have taken aR = 0.06 Å and ac = 0.68 Å, as in ref 60.
Figure 5. Calculated potential energy curves for NaHe2000. Note the different vertical scales. Solid line: Σ states (m = 0); dashed: Π states (m = 1); dotted: Δ states (m = 2). The labels at large zA indicate the atomic asymptotic states.
For a given m, the matrix elements of M , a function of the dropletalkali distance zA, are integrals of the type Z Ylm ðθ, jÞYλm ðθ, jÞΨnl ðr 0 ÞVeHe ðF0 Þ M nl, νλ ðzA Þ ¼ Ψνλ ðr 0 Þ dx dy dz
ð10Þ
with θ = arctan[(x þ y ) /(z zA)] and r0 as that for eq 8. Note that because the Ψ are real, all quantities in the integrand are real except for the factors eimj,eimj, which cancel out. Thus, the matrix is real and symmetric, and the azimuthal angle j plays no role in the calculation, as expected from the cylindrical symmetry of the problem. 2
2 1/2
’ WAVE FUNCTIONS AND POTENTIAL ENERGY CURVES The total matrix to be diagonalized (here shown with the energy ordering of the m = 0 block for the case of Na; 3s ground state) has the form 0
M 30, 30 þ E30 B B M 30, 31 E0 ¼ B B l @ l
M 30, 31 M 31, 31 þ E31 M 31, 40
333 M 31, 40 M 40, 40 þ E40
1 333C C C C A 3 33
ð11Þ Let us call E0nl(zA) the eigenvalues of E 0 (although n and l are no longer good quantum numbers, a one-to-one correspondence 6793
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Figure 6. Calculated potential energy curves for KHe2000. Solid line: Σ states (m = 0); dashed: Π states (m = 1); dotted: Δ states (m = 2). Note the different vertical scales. The labels at large zA indicate the atomic asymptotic states.
Figure 7. Calculated potential energy curves for RbHe2000. Note the different vertical scales. Solid line: Σ states (m = 0); dashed: Π states (m = 1); dotted: Δ states (m = 2). The labels at large zA indicate the atomic asymptotic states.
with Enl can be established via the asymptotic behavior of the adiabatic potential energy curves) and E00(zA) the lowest one (E030 for Na, E040 for K, etc.); assuming that the coredroplet interaction is the same for all excited states, the full potential curves being sought are calculated as V0(zA) þ E0n(zA) E00(zA). With this definition, the zero of the energy corresponds to the ground state of the bare atom; also, for the lowest-energy eigenvalue, the expression correctly simplifies to V0(zA), the potential energy curve of the pseudodiatomic ground state, defined in eq 8. The eigenvectors Ψ0nl = ∑νλ Unl,νλΨνλ of this matrix are the wave functions of the alkali atom as perturbed by the He droplet. Because the eigenvectors are an explicit function of the unperturbed wave functions Ψnl quantities such as permanent dipoles, electricdipole transition moments, or the change of the hyperfine constant24 can be immediately calculated from the calculated mixing coefficients Unl,νλ and basis functions 61 Ψnl (as done by Pascale in alkalirare gas systems) optionally supplemented with experimental values and with scaling laws for Rydberg atoms as needed [in refs 23 and 24, where we are interested in the value of |Ψ0n0(0)|2 in the ground electronic state due to Pauli forces alone (because we separately add the contribution due to van der Waals forces), we only use the kinetic energy term in eq 9, that is, we set VeHe(F) = (p2k20)/(2me)]. The potential energy curves are shown in Figures 58. As we show in the companion paper37 and in ref 6, the FranckCondon factors between the v = 0 vibrational state in the ground electronic state and that in dipole-allowed excited states reproduce the experimentally observed spectra
well enough to allow an unambiguous assignment. A better agreement may be obtained by refining the FranckCondon factor approximation (i.e., by accounting for the dependence of the transition dipole moment on zA) and by including neglected polarization effects in the calculation of the elements of the perturbation matrix. Qualitatively, one can notice that because of the increasing penetration of the valence electron into the droplet, excited states are increasingly repulsive at atomdroplet distances corresponding to the equilibrium distance of the system in its ground state (zA = 3133 Å), as reflected experimentally in increasingly blue-shifted spectra. The trend reverses for some of the highly excited states; several potential energy curves become increasingly flat, and the experimentally observed transition becomes correspondingly sharp. This corresponds experimentally to the situation where the valence electron is so diffuse that it interacts only minimally with the droplet and is to be expected in our model too for the same reason; because our model does not explicitly account for the polarization energy gain from the interaction of the unshielded alkali metal atom core with the droplet, the curves will always be slightly repulsive, or at most flat. Experimental observations with Na do show, in contrast, that energy levels can become lower than those of the free atom and, in particular, that polarization effects do lower the ionization energy of the alkali metal atom by an amount well reproduced by a continuous dielectric model.59 We plan to incorporate a similar correction in a future evolution of the model presented here. 6794
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positively charged core of the dopant atom. Barring a brute-force three-body calculation, which is computationally expensive, this effect can probably be accounted for by developing a reasonable interpolation scheme between the two known extreme cases, dopant atom in the ground state (negligible correction) and dopant atom in a highly excited state (AþHeN interaction). Explicit knowledge of a basis of wave functions for the valence electron of the dopant atom, and of their weight in the sum that defines the perturbed wave functions, allows the brute-force calculation of several quantities of interest (not presented here), such as change of hyperfine constant,24 permanent dipoles, and transition dipoles.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Present Addresses ^
Sincrotrone Trieste, Strada Statale 14 - km 163.5, 34149 Basovizza, Trieste, Italy.
Figure 8. Calculated potential energy curves for CsHe2000. Note the different vertical scales. Solid line: Σ states (m = 0); dashed: Π states (m = 1); dotted: Δ states (m = 2). The labels at large zA indicate the atomic asymptotic states.
’ CONCLUSIONS AND SUMMARY We have proposed a method for the calculation of potential energy curves and related electronic excitation spectra of alkali metal atoms captured in/on He nanodroplets that goes beyond the integral of pair potentials approach used so far. The method requires knowledge of several quantities, which can however be calculated in separate, relatively standard steps and combined only at the end into a perturbative approach. The following steps are necessary: (1) calculation of a basis set {Ψlm} of valence electron wave functions for the bare alkali metal atom; (2) calculation of the nanodroplet density F0 at equilibrium with the dopant alkali metal atom; (3) calculation for all dopant droplet distances zA of interest of the perturbation matrix elements due to the electronHe interaction in the basis {Ψlm} given the density F0; and (4) diagonalization of the energy matrix as a function of zA. Excitation spectra are calculated as the FranckCondon factors between the v = 0 vibrational state in the ground electronic state, and the excited state(s) of interest. The method produces potential energy curves and related excitation spectra that agree well with available experimental data, even for highly excited states where the more traditional approach fails. In its present formulation, the method is computationally affordable, even for the nonspecialist, but is clearly limited by the fact that polarization of the He by the alkali metal atom core has not been explicitly included; this limit is more evident for highly excited states where the more diffuse valence electron is less efficient at shielding the droplet from the
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’ NOTE ADDED AFTER ASAP PUBLICATION This article posted ASAP on March 24, 2011. Reference 6 has been revised. The correct version posted on March 29, 2011.
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