Solvation and spectral line shifts of chromium atoms in helium droplets based on a density functional theory approach.

The interaction of an electronically excited, single chromium (Cr) atom with superfluid helium nanodroplets of various size (10 to 2000 helium (He) atoms) is studied with helium density functional theory. Solvation energies and pseudo-diatomic potential energy surfaces are determined for Cr in its ground state as well as in the y(7)P, a(5)S, and y(5)P excited states. The necessary Cr-He pair potentials are calculated by standard methods of molecular orbital-based electronic structure theory. In its electronic ground state the Cr atom is found to be fully submerged in the droplet. A solvation shell structure is derived from fluctuations in the radial helium density. Electronic excitations of an embedded Cr atom are simulated by confronting the relaxed helium density (ρHe), obtained for Cr in the ground state, with interaction pair potentials of excited states. The resulting energy shifts for the transitions z(7)P ← a(7)S, y(7)P ← a(7)S, z(5)P ← a(5)S, and y(5)P ← a(5)S are compared to recent fluorescence and photoionization experiments.

Introduction

Chromium, a very versatile transition metal due to its half filled 3d and 4s shells, is found in 9 different oxidation states, ranging from −2 to 6. Its ground state is of septet multiplicity (a7S, [Ar]3d54s), with an outstanding magnetic moment of 6 μB. While Cr clusters are mostly formed in antiferromagnetic states,[1,2] the principal possibility to create high-spin nanoparticles[3] makes them a technologically interesting target. In this context, the experimental methods of helium nanodroplet isolation spectroscopy[4] offer a 2-fold advantage for studying cluster formation and manipulation. First, it has been shown that cluster formation on helium nanodroplets (HeN) is spin selective, with a clear preference for weakly bound high spin states.[3,5−8] Second, single species of the formed clusters are made spectroscopically accessible at very low temperature due to the cold He environment.[4,9] Recent experimental investigations of Cr on HeN in our group were based on mass spectroscopy,[10] followed by photoionization and fluorescence studies.[11−13] The particularly complex interaction between Cr and the He environment, with Cr turning out to be a borderline species in terms of its actual position on the droplet, triggered this theoretical study.

We investigate the influence of the HeN environment on Cr in its ground state and selected electronically excited states of the septet and quintet spin manifold by means of helium density functional theory (DFT). A powerful alternative to this approach are quantum Monte Carlo methods,[14] as has recently been shown for the 3Σu high spin state of Rb2.[15,16] However, though more qualitative in its analysis, the DFT approach provides us with a simple model of electronic excitation that allows us to interpret and understand basic spectroscopic features of the Cr–HeN system. Similar studies (ground states only) have been published recently for the Rb–HeN and Xe–Rb–HeN system.[17,18]

Computational Methods

Helium density functional theory is applied to calculate free energies and density profiles for Cr-doped helium nanodroplets consisting of up to 2000 He atoms. Note that, in contrast to the common DFT methods of electronic structure theory, the energy is not written as a functional of the electronic but of the atomic density.[19] Two functionals found application in the helium droplet community. The Orsay–Paris functional was proposed in refs (20−23) for both He isotopes. Our choice, the Orsay–Trento functional, was specifically tailored for isotopically homogeneous 4He systems, improving the description of static and dynamic properties of liquid 4He.[24]

In our computational approach the free energy of a doped He droplet is minimized for a given dopant position. The free energy F[ρ], a functional of the helium density ρ, may be written aswith E[ρ] as the Orsay–Trento density functional and Uext[ρ] as an external potential, which describes the interaction between the droplet and the dopant. Two constraints, the conservation of the particle number (N) and the center of mass (R⃗), appear as terms on the right-hand side, together with their corresponding Lagrange parameters, the chemical potential μ and the retaining force F⃗, respectively. The retaining force F⃗ fixes the distance between the center of mass of the HeN and the dopant to a predetermined value enabling the calculation of the Cr–HeN potential energy surface (PES). The DFT code we use was developed by Dalfovo and expanded by Lehmann and Schmied.[25,26] More recent developments in the field are a fully variational DFT,[27] which provides an improved description of strongly interacting dopants and the liquid–solid transition, and a time-dependent DFT code, which allows dynamical treatments of He systems.[28]

Ab Initio Calculations for the Cr–He Diatomic Molecule

Any interaction between the chromium atom in a certain electronic state and the helium droplet is represented by Uext[ρ] in eq 1. This term contains a density-weighted summation over Cr–He pair energy contributions. The latter can be obtained from diatomic Cr–He potential energy curves, with Cr in the relevant electronic state. The wanted Cr–He potential curves for the X7Σ+ ground state (a7S) and the lowest state in the quintet spin manifold, 15Σ+ (a5S), were taken from ref (29). Both are corrected for relativistic effects and finiteness of the basis set. The remaining excited states, 27Σ+/17Π, 47Σ+/47Π, 55Σ+/55Π, and 65Σ+/65Π, which correspond to the atomic excitations z7P, y7P, z5P, and y5P, were calculated with the Molpro quantum chemistry package[30] according to the following procedure: A state-averaged complete active space multiconfigurational selfconsistent field calculation (CASSCF)[33,34] was performed with the aug-cc-pVTZ-DK[31,32] basis set, followed by second order multireference configuration interaction calculation (MRCI).[33] (The Π-states in the septet-multiplicity were unsuccessfully treated with coupled cluster (CC) methods. The open shell character of Cr prohibits a reliable treatment with CC methods as shown in ref (29).)

DIM Approach

The interaction between HeN and the dopant appears as Uext[ρ] in eq 1. In our approach, this external potential is approximated via a density-weighted integration of Cr–He pair potential energies over space:

We set the origin of our coordinate system to the dopant position. For a HeN-centered atom in an electronic S state the choice of spherical coordinates is convenient since the integrand does not have any angular dependence, which reduces the problem to an integration over the radial component only. However, a slight displacement of the dopant from the droplet center immediately results in a breakdown of spherical symmetry, and the problem is best treated in cylindrical coordinates. Note that in this case the pair potential Vpair is still spherically symmetric (but the density ρ(r⃗) is not) and that a single coordinate is sufficient to describe the interaction. This situation changes if we allow the dopant atom to be in electronically excited states with angular momenta larger than zero. Now, the pair potential itself varies in two coordinates and needs to be integrated in two dimensions. For convenience, we choose a definition according to the illustration in Figure 1, with r as the distance between the Cr atom and a source point of He density, and Θ as the angle between vector r⃗ and the internuclear axis in a pseudo-diatomic picture.[35] This axis is defined as the vector between the dopant position and the center of mass calculated for a given helium density distribution.

Figure 1

Coordinates for the integration over the He–Cr pair potential for electronic states with angular momenta larger than zero.

Finally, the actual two-dimensional interaction potential Vpair needs to be derived for each value of Λ, the projection of the angular momentum onto the internuclear axis. Here we follow the diatomics in molecules (DIM) approach first proposed by Ellison in refs (36) and (37), by now a well-established method within the helium droplet community.[28,38−42] In our concrete case of atomic P excitations, the necessary Cr–He Σ and Π states are represented by ab initio PESs VΣ(r) and VΠ(r) obtained from the Molpro quantum chemistry package. They can be combined to form on-droplet Σ′ and Π′ states, expressed by two two-dimensional PESs VΣ′(r, Θ) and VΠ′(r, Θ):

A constant factor stemming from the formal integration over the azimuth φ has already been adsorbed in these equations. According to eq 2, these PESs need to be integrated over spatial coordinates for the final evaluation of Uext[ρ]. This is most easily done in cylindrical coordinates, with the interatomic axis as the z axis in the new coordinate system. Note that the coordinate transformationsgive rise to a pole in the integrand for arbitrary He densities. However, since the He density is zero at the dopant position by necessity, this point can be excluded.

Frozen Droplet: An Approximation to Atomic Excitations

A series of DFT calculations, performed as described above, but repeated for different interatomic distances between droplet and Cr atom, allows us to determine the preferred localization of the dopant for a given interaction potential. An effective Cr–HeN PES is obtained, from which we derive the solvation energy as the difference between free energies of the doped and the undoped droplet. We refer to this type of calculation as relaxed since the He density is fully adapted to effects introduced by Uext, the interaction potential of the dopant.

In contrast to these relaxation calculations, “frozen” droplet calculations are single-point energy evaluations without readjustment of the He density, although the interaction potential might have changed. Such an evaluation can be interpreted as a zeroth-order approach to the electronic excitation of a Cr atom on HeN: The optical excitation of the dopant is fast compared to the relaxation of the droplet.[28,43] Therefore, the HeN density distribution can be assumed to be constant during excitation. To calculate the shifts in the transition energy for a certain electronic excitation from atomic state A to B, we let the doped HeN relax in state A at first. The obtained density is then used to evaluate the energies for states B and A, and their energetic difference is interpreted as on-droplet excitation energy.

Line broadening effects have been extensively studied in refs (39) and (40). Two contributions can be identified: The thermal motion of the dopant within the HeN matrix results in a minor line broadening, but oscillations of the HeN bubble around the dopant lead to additional shifts and line-broadening especially for submerged dopants. A larger line-broadening effect is caused by the spherically symmetric breathing mode.[40] We limit our investigation to the droplet-centered atomic S state a7S and spherical symmetry in the initial state and follow a simplified approach of ref (40). We start with the ground state equilibrium density distribution of the doped HeN ρ(r). The breathing mode bubble oscillation is implemented by applying a displacement α on the density distribution including a renormalization: Both the energy of the ground state Uext,GS[ρ′(r, α)] and the excited state Uext,ES[ρ′(r, α)] were calculated within the frozen droplet approach according to eq 2. The transition energies for specific bubble deformations (α) are then obtained bywith Em,A being the energy of the atomic transition with the identifier m, which is not taken into account in eq 2. In order to calculate the probability for displacements α, the bubble oscillation is described by a harmonic oscillator with the wave function ψ0 (r), the hydrodynamic mass M0*, the angular frequency ω0, and the equilibrium bubble radius R0. For the calculation of the broadening the potentials are assumed to be of Lennard-Jones type (rmin, ϵmin) resulting in the following equations:

The density[19,44] ρHe,b = 0.0218 Å–3 and surface tension[45] σHe = 0.179 cm–1 Å–2 of bulk liquid He are used. The Ancilotto parameter λA is discussed in section 2.4, and its value can be found in Table 1. With these quantities, the energy ℏω0 and the hydrodynamic mass M0* of the breathing mode can be determined:

Table 1

Ancilotto (λA) and de Boer (λB) Parameters Obtained for Selected States of Diatomic Cr–He

lim.	term	rmin	εd	λB	λA
a 7S	X 7Σ+ a	5.038	4.88	0.21	2.69
	X 7Σ+	5.145	4.72	0.23	2.41
z 7P°	2 7Σ+	7.781	1.02	0.42	0.87
	1 7Π	1.983	672.67	0.01	146.20
y 7P°	4 7Σ+	7.575	1.34	0.34	1.11
	4 7Π	6.399	2.50	0.25	1.75
a 5S	1 5Σ+	6.067	1,48	0.47	0.98
z 5P°	5 5Σ+	8.067	0.86	0.46	0.76
	5 5Π	3.041	79.67	0.04	26.56
y 5P°	6 5Σ+	6.668	1.99	0.29	1.46
	6 5Π	6.408	2.28	0.28	1.60

Calculated at the CCSD(T) level. All others are based on MRCI results.

The following wave function can then be applied for the breathing deformation:

A combination of eq 5 and 11 allows to calculate the spectrum:

Comparison to Basic Descriptors

Two basic descriptors, the Ancilotto parameter λA and the de Boer factor λB, are sometimes applied to obtain first predictions for the interaction of a certain dopant with HeN.[46] The only input needed for this estimation are basic parameters of the corresponding He–dopant diatomic PES. λA is a dimensionless parameter, formed by the ratio of the free energy gain due to the dopant–matrix interaction and the free energy cost due to the formation of a bubble around the dopant. The dopant–matrix interaction is based on a diatomic PES with spherical symmetry. Therefore, λA relates the PES properties, the well depth εd and the equilibrium separation rmin, the number density of the matrix particles ρnum, and the surface tension σ as follows:

For liquid 4He a threshold value of λA = 1.9 has been found, distinguishing between surface residing (λA < 1.9) and submerging (λA > 1.9) species. Simplifications of λA are the assumption of spherically symmetric and idealized Lennard–Jones-type interaction and the neglect of quantum effects such as zero point energy. The latter can be estimated via the de Boer parameter[46] (λB), which relates εd, rmin, the dopant mass m, and Planck’s constant h as follows:

Typical values are λB > 1 for light atoms (H–He, Li–He, and Na–He) with a significant zero-point energy and λB < 0.15 for heavy atoms and molecules (Ar–He and SF6–He) where quantum mechanical effects only play a minor role.

Results and Discussion

We focus on a theoretical description of the interaction between a single Cr atom and He2000. Besides the a7S ground state of the atom, five electronic excitations of the Cr atom (z7P, y7P, a5S, z5P, and y5P) are modeled and compared to recent experimental results of our group.[11−13] This section is divided into 6 subsections. We start with the discussion of the diatomic Cr–He PES[29] as basic requirements for the calculations. Second, we address the localization of the Cr on HeN in different electronic states and compare our DFT results to qualitative predictions based on the Ancilotto parameter. Third, the line shifts on optical excitation are calculated and their droplet size dependency is shown. Fourth, line broadening is discussed. Fifth, the size of the first solvation shell around ground state Cr (a7S) is deduced. In the last subsection we compare our results to experimental data where possible.

Diatomic Cr–He Potential Energy Surfaces

The Cr–He PES for the X7Σ+ ground state (atomic a7S state) and the 15Σ+ lowest quintet state (atomic a5S) are taken from ref (29). In both cases, a Complete Active Space SCF (CASSCF) calculation was followed by a Davidson-corrected multireference calculation of second order (MRCI-Q). For the ground state, a powerful single-reference method (coupled cluster with singles, doubles, and perturbative triples (CCSD(T)))[47] could be applied as well for direct comparison. The resulting three PESs are plotted in Figure 2. All curves have been corrected for relativistic effects and extrapolated to the basis set limit.

Figure 2

CCSD(T) and MRCI results for the PESs of Cr–He in the X7Σ+ ground state and the 15Σ+ lowest quintet state.

Although several electronically excited Cr–He PESs were available from ref (29), we decided to recalculate all of them together with the missing higher excitations in the quintet manifold. The inclusion of more excited states was only feasible at the cost of using a smaller basis set, in our case aug-cc-pVTZ-DK,[31,32] which necessitated the recalculation of all states for a more balanced description of both spin manifolds.

Figure 3 shows the PESs obtained for diatomic states with z7P atomic limit (27Σ+, 17Π) and with z5P atomic limit (55Σ+,55Π). Note the big difference between Σ and Π PESs: 17Π and 55Π show a remarkably deep potential minimum, while 27Σ+ and the 55Σ+ do not. A similar behavior has been found for other species like copper, silver, and gold.[48−50] In contrast, states with y7P atomic limit (47Σ+, 47Π) and with y5P atomic limit (65Σ+, 65Π) show the typical shallow minimum as can be seen from Figure 4.

Figure 3

Comparison of Σ and Π states corresponding to z7P and z5P. Note the pronounced minima for 17Π and 55Π.

Figure 4

PESs for the excited Cr–He states 47Σ+/47Π (y7P) and 65Σ+/65Π (y5P).

The excited Cr–He states, 27Σ+/17Π (z7P), 47Σ+/47Π (y7P), 55Σ+/55Π (z5P), and 65Σ+/65Π (y5P), are mixed in pairs by the DIM approach to on-droplet Σ′ and Π′ states. Since the quintet and septet states show similar behavior, only the on-droplet septet states are shown in Figure 5. Because of the enormous difference in well depth between the shallow 27Σ+ and the deep 17Π PESs (z7P), a cutoff at −3.5 cm–1 was chosen for the sake of illustration. The actual three-dimensional PES can be obtained by rotating the two-dimensional figures around the z axis. The Σ′ PESs have a pronounced toroidal well around the Cr atom, in combination with the spherical shell of a local PES minimum. The Π′ PES shows a similar spherical shell of local minimum but has pronounced global maxima on either side of the dopant along the z axis. Considering the on-droplet states corresponding to the z7P and z5P states it seems possible that exciplexes can be formed by trapping He atoms in the deep potential minima.[50]

Figure 5

Contour plots of Σ′ and Π′ and the on-droplet PESs obtained from the DIM method for the 27Σ+/17Π states (upper row) and 47Σ+/47Π states (lower row), as a function of z and ρzyl for arbitrary φ. Energies are cut off at −3.5 cm–1 for clarity.

Localization of Cr on HeN

We start with the calculation of Ancilotto parameters (λA) for the diatomic Cr–He PES as discussed in section 3.1. They are shown in Table 1, together with the essential parameters of the PES (minimum εd and equilibrium distance rmin) and the solvation de Boer factor (λB). The calculated λA indicate a submerged position of Cr–He in the X7Σ+ (a7S) ground state and the strongly bound Π states, 17Π (z7P) and 55Π (z5P). All other states show a surface residing position of the dopant. However, considering elements with similar diatomic potential energy surfaces, such as Mg,[40,51−54] the behavior of Cr on HeN cannot be determined reliably. Therefore, we switch to the DFT approach for further investigation. In Figure 6, the calculated HeN density distributions of ground state Cr centered in HeN for N ranging from 10 to 2000 are compared to the undoped He2000 and bulk liquid He, using the ground state interaction potential for this study. Density peaks close to the dopant indicate a shell structure, which we discuss in section 3.5.

Figure 6

Density distributions for different HeN sizes doped with Cr in X7Σ′ (a7S) show a shell structure. Especially, a first and a second solvation shell at approximately 5.5 and 8.5 Å, respectively, are visible. Bulk density values are reached already for N = 500.

The density of the undoped He2000 is almost identical with the bulk liquid He density up to a radius of 21.5 Å, but then drops with small oscillations over a length of 10.5 Å to zero. In the case of the doped He2000, the Cr atom pushes the He away and forms a bubble (zero He density) with a radius of 3.6 Å. A steep increase in He density can be seen near the rim of this bubble, with a maximum of 160% of bulk He density at 5.2 Å. This density peak aligns well with the minimum of the Cr–He PES (5.0 Å). It is followed by decaying density oscillations, but after the fourth density maximum at 16.4 Å the oscillations become barely visible at the given scale. From there, the doped and undoped He2000 density distributions are essentially the same.

For quantitative analysis, the Cr–He2000 PES is investigated for the states X7Σ′ (a7S) and 47Σ′/47Π′ (y7P), see Figure 7, and 15Σ′ (a5S) and 65Σ′/65Π′ (y5P), see Figure 8. The Cr–He2000 PES was determined by minimizing the energy functional in eq 1, including only the first and second term and subtracting the energy of the empty He2000. On the basis of these Cr–He2000 PESs, the localization of the dopant can be seen by minimization of the solvation energy. Starting with the Cr–He2000 ground state X7Σ′ (a7S), we compare the effect of different Cr–He PES, CCSD(T), and MRCI-Q, on the Cr–He2000 system. These PESs have their minimum in solvation energy of −52.9 and −41.0 cm–1 in the center of the droplet for the CCSD(T) and MRCI-Q Cr–He PESs, respectively. Since the Cr–He ground state is treated more accurately by CCSD(T) than MRCI-Q,[29] the CCSD(T) results should be favored. Both Cr–He2000 PES show a typical behavior of submerging dopants. The PES is nearly flat inside the droplet, with the inner 45% of the droplet radius showing an energy increase of at most 1.0%. Reaching the droplet radius of 28 Å, the solvation energy increases by 30%. The solvation energy approaches zero at ∼12 Å outside the droplet radius. This slow increase in solvation energy is associated with the abating He density around the droplet radius.

Figure 7

Cr–He2000 PESs for the X7Σ′ (a7S) and excited 47Σ′/47Π′ (y7P) states. X7Σ′: comparison of CCSD(T) and MRCI-Q PESs from Figure 2; Cr resides inside HeN. 47Σ′/47Π′: a surface residing position of the dopant is favored. A value of zero is approached at large radii, the energy of the relaxed He2000 without interaction with Cr.

Figure 8

Cr–HeN PESs of the 15Σ′ (a5S) and 65Σ′/65Π′ (y5P) states; all states show a localization of the dopant close to the surface. Zero point of energy marks the empty He2000.

Two on-droplet excited states, 47Σ′/47Π′ (y7P), are depicted in the same figure (Figure 7). In contrast to the ground state X7Σ′ (a7S), these excited states favor a surface residing position (47Σ′, −13.7 cm–1 at 28 Å; 47Π′, −16.8 cm–1 at 30 Å). This is indicated by a global minimum of the Cr–He2000 PES close to the surface and positive solvation energies upon full submersion of the dopant. However, a stable surface residing position can only be obtained if the excess energy, due to an on-droplet position of the Cr atom on excitation, can be dissipated into HeN without destroying it.

In the quintet manifold (see Figure 8), the Cr–He2000 PES of the lowest lying state 15Σ′ (a5S) indicates a surface residing position (−7.2 cm–1 at 31 Å). The 65Σ′/65Π′ (y5P) PES shows the interesting feature of a double minimum and are practically identical within less than 1 cm–1. The local minimum (∼−9 cm–1) is found in the droplet center. A tiny barrier of less than 1 cm–1 separates it from the global minimum (∼−20 cm–1), which occurs close to the droplet surface at about 28 Å.

Line Shifts in Optical Excitation Spectra

The frozen droplet approximation was applied to calculate shifts in the transition energies for the z7P ← a7S, y7P ← a7S, z5P ← a5S, and y5P ← a5S absorption bands. Starting from the on-droplet ground state X7Σ′ (a7S), the Cr atom is found within the HeN. An excitation into the 27Σ′/17Π′ (z7P) states requires an excess energy of 340 cm–1 for a vertical excitation within the DIM picture. The excitation into the 47Σ′/47Π′ (y7P) shows an even more distinct blue shift of 390 cm–1. Figure 9 shows the dependency of this energy shift on the position of the Cr on He2000 for the four septet states. For smallest deviations from the center position, the spherical symmetry of the bubble is almost conserved, leading to identical shifts for excited Σ′ and Π′ potentials. With increasing distance, their difference in shape becomes more and more important and leads to a split in energy. Small steps in the transition energy shift occur due to the interference of HeN density fluctuations with the interaction potential.

Figure 9

Transition energy shifts in the septet manifold: 27Σ′/17Π′ ← X7Σ′ (z7P ← a7S) and 47Σ′/47Π′ ← X7Σ′ (y7P ← a7S); the vertical transition region is marked in gray. A blueshift of 340 and 390 cm–1 is expected. The zero-line is the energy of the free-atom transition.

A different scenario is found in the quintet system, where the initial state is the surface residing 15Σ′ (a5S) state. The shifts of the transition into the four quintet states 57Σ′/57Π′ (z5P) and 67Σ′/67Π′ (y5P) are shown in Figure 10.

Figure 10

Transition energy shifts in the quintet manifold: the Cr atom is surface residing at around 33 Å (the vertical transition region is marked as a gray bar). A blueshift of 38 and 2 cm–1 is expected for the 55Σ′/55Π′ ← 15Σ′ (z5P ← a5S) transitions. A redshift of 7 and 9 cm–1 is found for the 65Σ′/65Π′ ← 15Σ′ (y5P ← a5S). Zero point of energy marks the free Cr atom.

A description in the pseudo-diatomic picture is appropriate because of the out-of-center location. At 31 Å, the equilibrium position of the 15Σ′ (a5S) state, the transition into the 55Σ′ state, is blueshifted by 38 cm–1. A minimal blueshift of 2 cm–1 is found for the 55Π′ ← 15Σ′ transition. The 65Σ′/65Π′ (y5P) states, however, show a slight redshift of 7 and 9 cm–1, respectively. Their potential well is deeper and occurs at slightly larger equilibrium distance than 15Σ′ (a5S).

Since a HeN size distribution is found in experiments rather than a single HeN size, the droplet-size dependency of this blueshift was studied in the septet manifold, see Figure 11. The Cr atom was placed in the center of HeN with 6 to 105 He atoms, and the vertical transition from the X7Σ′ (a7S) state into the 27Σ′/17Π′ (z7P) and 47Σ′/47Π′ (y7P) states was investigated. Because of the centered Cr atom, the on-droplet Σ′ and Π′ laser transitions are completely indistinguishable. The blueshift rises steeply for small HeN but remains fairly constant at 335(10) cm–1 (z7P) and 385(10) cm–1 (y7P) for HeN with N > 200. However, droplet sizes in experiments are usually considerably larger than 200 He atoms.[11−13]

Figure 11

Droplet size dependence of the transition energy shift for the two transitions 27Σ′/17Π′ ← X7Σ′ (z7P ← a7S) and 47Σ′/47Π′ ← X7Σ′ (y7P ← a7S) in the frozen droplet approximation with the Cr atom in the droplet center. The blueshift on excitation remains fairly constant for N > 200.

The interaction of Cr with small HeN, including 6 (He6) and 12 (He12) He atoms, was also investigated based on CCSD(T) calculations with aug-cc-pVTZ-DK basis sets including DK correction. The interaction between the He atoms is included on a quantum mechanical level as well as the Cr–He interaction. This is a difference to the DFT approach where the Cr–He interaction is only considered by a diatomic potential. The calculation therefore includes many body interactions between Cr atom and several He atoms. The He atoms were fixed to the corners of a regular octahedron (He6) and a regular icosahedron (He12). The energy was calculated for various radii of the octahedron and the icosahedron resulting in the PESs shown in Figure 12. The minima of these Cr–He6 and Cr–He12 PESs are found at −19.4 and −39.2 cm–1, respectively, and the equilibrium radii lie in the range between 5.3 to 5.5 Å for both PESs. In comparison, the DFT calculations show a solvation energy of −14.7 cm–1 for Cr–He6 and −30.5 cm–1 for Cr–He12. The equilibrium distances in Figure 11 (5.5 Å for both systems) are in good agreement with the position of the first density peak (see Figure 14a).

Figure 12

Cr–He6 (octahedron) and Cr–He12 (icosahedron) PESs calculated with CCSD(T) based on aug-cc-pVTZ-DK basis sets; r is the distance between the centered Cr atom and the He atoms.

Figure 14

(a) Density profiles of He10 to He14; (b) density profiles are normalized to their atom number, and the difference between the ascending droplet sizes are formed. The peak emerging at 8.3 Å indicates a first solvation shell of 12 He atoms.

Line Broadening

Besides the blueshift, also line broadening is observed in experiments.[11−13] A line broadening due to the droplet-size dependency of the transition energy can usually be neglected. The experimental droplet-size distributions are peaked around 2000.[11] In this range, the size dependency of the blueshift is negligible, as can be seen in Figure 11. Thermal motion of the dopant within the HeN, also gives no significant contribution to line broadening. The wave function of Cr within He2000 extends to about ∼10 Å for the droplet temperature. There is only a small change in transition energy for such a displacement, see gray area in Figure 9. A large contribution to line broadening can be expected from deformations and oscillations of the bubble in HeN around the dopant. The breathing mode of the HeN bubble should dominate the line broadening[40] and was considered by the approach outlined in section 2.3. On the basis of these calculations, the hydrodynamic mass and the ground state energy of the breathing mode is found to be M0* = ∼99 u and ω0 = ∼14.2 cm–1, respectively. In Figure 13, the spectra computed with eq 12 is compared to experimental values taken from ref (12). The breathing mode bubble oscillations lead to a line broadening of ∼200 cm–1 for the 47Σ′/47Π′ ← X7Σ′ (y7P ← a7S) transition. Spin–orbit splitting and relative intensities are taken from the NIST database[55] and combined with our results for improved accuracy. This consideration leads to an even higher line broadening of about ∼350 cm–1 with a maximum at ∼28 200 cm–1. Compared to the atomic line center at 27 847.78 cm–1, a shift of ∼350 cm–1 is found. The overestimation of the line shift may be caused by neglecting the quadrupole deformation, see ref (40).

Figure 13

Comparing the experimental (LIF[12]) and theoretical spectra (SUM) for the 47Σ′/47Π′ ← X7Σ′ (y7P ← a7S) transition; for the theoretical spectra, the spin–orbit splitting was included by taking the atomic values from NIST database into account.[55]

Solvation Shell

Density profiles of the HeN show a distinct shell-like structure around Cr dopants, see Figure 6. The first solvation shell is indicated by the global density maximum at 5.5 Å and a second solvation shell emerges at ∼8.3 Å. To deduce the number of the atoms in the first solvation shell, first the depicted density distributions of He10 to He14 around a Cr atom were calculated (Figure 14a). The size of the first solvation shell can be determined by adding He atoms one by one and watching how the normalized distribution ρ(N)/N around the dopant evolves. Figure 13b shows a plot of the differences of normalized density distributionsversus the coordinate r. Up to N = 12, Δρ is negative at ∼8.3 Å, but beyond N = 13, the normalized density starts to rise, which indicates the beginning of a second shell formation at this position. The population of the second solvation shell, therefore, starts after 12 He atoms are accumulated in the first shell.

Comparison to Experiment

Experimentally, a band shift of about 300 cm–1 to the blue is found[11−13] for the transitions z7P ← a7S and y7P ← a7S. This is in good agreement with our theoretical values of 340 cm–1 (z7P ← a7S) and 390 cm–1 (y7P ← a7S). The line broadening of the y7P ← a7S transition was treated separately by including He bubble oscillations leading to a shift of ∼350 cm–1 and a broadening of ∼350 cm–1. After excitation into the states z7P and y7P, sharp lines are observed in successive photo ionization[11] and fluorescence spectra[12] and interpreted as resulting from ejected free Cr atoms. We can confirm this interpretation for the y7P, where the Cr atom is pushed to the HeN surface after excitation and may even leave the HeN. The probability of Cr captured in a surface residing position could not be determined and will depend on the distribution of excess energy after excitation into a number of different channels. For example, laser excitation from the Cr ground state into 47Σ′/47Π′ (y7P) in the “frozen He” approximation leaves the excited Cr atom with an excess energy of 390 – 8 = 382 cm–1, compared to the equilibrium situation in which the helium environment would have readjusted to the excited Cr orbital. Even in the equilibrium situation, the solvation energy would be positive and not favor an inside location. The excess energy has to be distributed into helium modes and kinetic energy of the Cr atom. In this case, it is unlikely that the atom arrives at the surface with an energy low enough to be captured in the potential minimum of approximately 15 cm–1 well depth. A different case may be observable if Cr in its metastable a5S state could be attached to a droplet and excited from the corresponding 15Σ′ state into 65Σ′/65Π′, which is also bound to the surface.

Conclusions

The interaction between Cr and HeN was investigated by density functional calculations for the He density[24] based on recently calculated diatomic Cr–He PESs.[29] The localization of the Cr atom on He2000 and expected shifts of transition energies due to Cr–HeN interaction were studied for the following atomic Cr states: a7S, z7P, y7P, a5S, z5P, and y5P. A droplet size dependency was investigated for the septet system. The solvation energy was calculated as a function of the Cr–HeN distance. On the basis of the ground state Cr–He2000 PES X7Σ′ (a7S), we showed that Cr resides inside HeN with a solvation energy of −53 cm–1. Because of the nonspherically symmetric Cr–HeN interaction of the excited states, the diatomics in molecules approach (DIM) was applied when required. The Cr–He2000 states 47Σ′ and 47Π′ (y7P) are surface residing with a binding energy of −14 and −17 cm–1 at an equilibrium distance of 28 and 30 Å, respectively. Transitions from the ground state X7Σ′ (a7S) into the excited septet states, 27Σ′/17Π′ (z7P) and 47Σ′/47Π′ (y7P), show a blueshift of 340 and 390 cm–1, respectively. A droplet size dependency of the blueshift was investigated but found only to be significant for droplet sizes up to 200 He atoms. Line broadening based on the breathing mode bubble oscillation was found to be ∼350 cm–1 for the 47Σ′/47Π′ ← X7Σ′ (y7P ← a7S) transition.

The shifts of transition energies were also investigated for quintet states. The lowest quintet state 15Σ′ (a5S) is surface residing at 31 Å and weakly bound with an energy of −7.2 cm–1. The excited states 65Σ′ and 65Π′ (y5P) are also found at the surface both at 28 Å with binding energies of −19 and −20 cm–1, respectively. Besides the surface location, also a local minimum of −9 cm–1 inside HeN was found for both states. The transition energy shifts starting from the lowest lying quintet state 15Σ′ are small compared to the septet system due to the surface residing position of the initial state.

Density profiles for the droplet-centered X7Σ′ (a7S) show a shell structure and a growth of the second solvation shell after 12 He atoms.