**2. Perturber effects on low-***n* **Rydberg states**

#### **2.1. Supercritical fluids**

A supercritical fluid (SCF) exists at a temperature higher than the critical temperature and, therefore, has properties of both a liquid and a gas [41]. For example, SCFs have the large compressibility characteristic of gases, but the potential to solvate materials like a liquid. Moreover, near the critical point the correlation length of perturber molecules becomes unbounded, which induces an increase in local fluid inhomogeneities [41].

The local density *ρ*(*r*) of a perturbing fluid around a central species (either the dopant or a single perturber) is defined by [42, 43]

$$
\rho(r) := \lg(r) \,\rho\,\,\,\,\,
$$

where *g*(*r*) is the radial distribution function and *ρ* is the bulk density. For neat fluids and for most dilute dopant/perturber systems, the interactions between the species in the system are attractive in nature. Thus, the perturber forms at least one solvent shell around the dopant (or a central perturber). As the dilute dopant/perturber system approaches the critical density and temperature of the perturber, the intermolecular interactions increase. This increase leads to a change in the behavior of the local density as a function of the bulk density [41]. Therefore, the changes in fluid properties near the critical point are due to the higher correlation between the species in the fluid. These intermolecular correlations are usually probed spectroscopically by dissolving a dopant molecule in the SCF. Since electronic excited states are incredibly sensitive to local fluid environment, fluctuations in the fluid environment can be investigated by monitoring variations in the absorption or emission of the dopant.

Fig. 1 gives three example spectroscopic studies of a dopant in near critical point SCFs. Fig 1a shows the energy position of an anthracene emission line for anthracene doped into near critical point carbon dioxide [44]. Unfortunately, no emission data on non-critical isotherms were measured in this fluorescence emission study. However, the maximum emission position shows a striking critical effect in comparison to a calculated baseline. A more complete investigation of temperature effects on the local density of SCFs via UV photoabsorption [45] of ethyl p-(N,N-dimethylamino)benzoate (DMAEB) in supercritical CHF3 is presented in Fig. 1b for three isotherms near the critical isotherm. The isotherms shown are at the reduced temperature *Tr* [where *Tr* ≡ *T*/*Tc* with *Tc* being the critical temperature of the fluid] of 1.01, 1.06 and 1.11. Although the three isotherms are evenly spaced, the photoabsorption shifts are very similar on the *Tr* = 1.06 and *Tr* = 1.11 isotherms, while the shift on the

*Tr* = 1.01 isotherm is significantly different. Therefore, the perturber induced shift is only temperature sensitive near the critical point of the perturber. Urdahl, *et al.* [46, 47] studied the W(CO)6 T1*<sup>u</sup>* asymmetric CO stretching mode doped into supercritical carbon dioxide, ethane and trifluoromethane by IR photoabsorption. All three systems show similar behavior, an example of which is presented in Fig. 1c. Again, the absorption band position changes significantly along the critical isotherm near the critical density. However, extracting the data presented in Fig. 1 is difficult. Moreover, most of the previous work [41] has used dopant valence transitions as probes, and these states tend to be less sensitive to local environments [1]. In the present work, we investigate near critical point SCFs using low-*n* Rydberg state transitions as the probe. Since Rydberg states are hydrogen-like, it should be possible to model these states within the statistical mechanical theory of spectral line-broadening.

**Figure 1.** (a) The energy of fluorescence emission for (•) anthracene doped into supercritical carbon dioxide at a reduced temperature *Tr* � 1.01 plotted as a function of carbon dioxide number density *ρ*CO2 . The dashed line (- - -) is a reference line calculated using perturber bulk densities. Adapted from [44]. (b) The perturber induced energy shifts of the photoabsorption maximum for DMAEB doped into supercritical CHF3 at (•) 30.0◦C [i.e., *Tr* � 1.01], () 44.7◦C, and () 59.6◦C, plotted as a function of CHF3 number density *ρ*CHF3 . Adapted from [45]. (c) The energy position of infrared absorption for the W(CO)6 CO stretching mode in supercritical carbon dioxide at (•) 33◦C [i.e., *Tr* � 1.01] and at () 50◦C, plotted as a function of *ρ*CO2 . Adapted from [46, 47]. The solid lines in (a) - (c) are provided as a visual aid.

#### **2.2. Theoretical model**

2 Will-be-set-by-IN-TECH

methodology for this extended study of dopant/perturber interactions are discussed in Section 3. Section 4 presents a review of our results for low-*n* atomic and molecular Rydberg states in atomic supercritical fluids. The accuracy of a semi-classical statistical line shape analysis is demonstrated, and the results are then used to obtain the perturber-induced energy shifts of the primary low-*n* Rydberg transitions. A striking critical point effect in this energy shift was observed for all of the dopant/perturber systems presented here. A discussion of the ways in which the dopant/perturber interactions influence the perturber-induced energy shift is also presented in Section 4. We conclude with an explanation of the importance of the inclusion of three-body interactions in the line-shape analysis, and with a discussion of how

this model changes when confronted with non-spherical perturbers and polar fluids.

unbounded, which induces an increase in local fluid inhomogeneities [41].

by monitoring variations in the absorption or emission of the dopant.

A supercritical fluid (SCF) exists at a temperature higher than the critical temperature and, therefore, has properties of both a liquid and a gas [41]. For example, SCFs have the large compressibility characteristic of gases, but the potential to solvate materials like a liquid. Moreover, near the critical point the correlation length of perturber molecules becomes

The local density *ρ*(*r*) of a perturbing fluid around a central species (either the dopant or a

*ρ*(*r*) = *g*(*r*) *ρ* ,

where *g*(*r*) is the radial distribution function and *ρ* is the bulk density. For neat fluids and for most dilute dopant/perturber systems, the interactions between the species in the system are attractive in nature. Thus, the perturber forms at least one solvent shell around the dopant (or a central perturber). As the dilute dopant/perturber system approaches the critical density and temperature of the perturber, the intermolecular interactions increase. This increase leads to a change in the behavior of the local density as a function of the bulk density [41]. Therefore, the changes in fluid properties near the critical point are due to the higher correlation between the species in the fluid. These intermolecular correlations are usually probed spectroscopically by dissolving a dopant molecule in the SCF. Since electronic excited states are incredibly sensitive to local fluid environment, fluctuations in the fluid environment can be investigated

Fig. 1 gives three example spectroscopic studies of a dopant in near critical point SCFs. Fig 1a shows the energy position of an anthracene emission line for anthracene doped into near critical point carbon dioxide [44]. Unfortunately, no emission data on non-critical isotherms were measured in this fluorescence emission study. However, the maximum emission position shows a striking critical effect in comparison to a calculated baseline. A more complete investigation of temperature effects on the local density of SCFs via UV photoabsorption [45] of ethyl p-(N,N-dimethylamino)benzoate (DMAEB) in supercritical CHF3 is presented in Fig. 1b for three isotherms near the critical isotherm. The isotherms shown are at the reduced temperature *Tr* [where *Tr* ≡ *T*/*Tc* with *Tc* being the critical temperature of the fluid] of 1.01, 1.06 and 1.11. Although the three isotherms are evenly spaced, the photoabsorption shifts are very similar on the *Tr* = 1.06 and *Tr* = 1.11 isotherms, while the shift on the

**2. Perturber effects on low-***n* **Rydberg states**

**2.1. Supercritical fluids**

single perturber) is defined by [42, 43]

Due to the hydrogenic properties of Rydberg states, the optical electron is in general insensitive to the structure of the cationic core. Therefore, both atomic and molecular Rydberg transitions can be modeled within the same theory. In a very dilute dopant/perturber system, assuming that the dopant Rydberg transition is at high energy [i.e. *β* (*Eg* − *Ee*) � 1, *β* = 1 / (*k T*)], the Schrödinger equation is

$$H|\Psi\rangle = E|\Psi\rangle,\tag{1}$$

where the eigenfunction |Ψ� is a product of the dopant electronic wavefunction |*α*� and the individual perturber wavefunctions |*ψi*; *α*�. The Hamiltonian *H* in eq. (1) is the sum of several individual Hamiltonians, namely the Hamiltonian for the free dopant, the Hamiltonian for the free perturber, and the Hamiltonian for the dopant/perturber intermolecular correlation. The Hamiltonian *H*FD for the free dopant is [48]

$$H\_{\rm FD} = \sum\_{\mathfrak{a}} E\_{\mathfrak{a}} |\mathfrak{a}\rangle\langle\mathfrak{a}| \,\, \!/ \mathfrak{a}\vert\, \!/ \,\, \tag{2}$$

where *α* = *e*, *g* with *e* and *g* representing the excited state and ground state of the free dopant. The Hamiltonian *H*FP for the free non-interacting perturber is given by [48]

$$H\_{\rm FF} = -\sum\_{i}^{N} \frac{\hbar^2}{2m} \nabla\_i^2 \tag{3}$$

where *N* is the total number of perturbing atoms in the range of the Rydberg optical electron. Finally, the Hamiltonian *H*PD for the intermolecular interaction is [48]

$$H\_{\rm FD} = \sum\_{\mathfrak{a}} \sum\_{i} \left[ V\_{\mathfrak{a}}(r\_i) \right.\\ \left. + V'(r\_i) \right] \left| \Psi \right> \langle \Psi \rangle \,\,\,\tag{4}$$

where *V*(*r*) is the dopant/perturber intermolecular potential, and *V*� (*r*) is the perturber/perturber intermolecular potential. Therefore, the total Hamiltonian H is [48]

$$H = H\_{\rm FD} + H\_{\rm FP} + H\_{\rm FD} \tag{5}$$

which can be rewritten as a ground state Hamiltonian expectation value *Hg* and an excited state Hamiltonian expectation value *He*.

The absorption coefficient function is given as the Fourier transform [48]

$$\mathfrak{L}(\omega) \equiv \frac{1}{2\pi} \int\_{-\infty}^{\infty} dt \,\mathrm{e}^{-i\omega t} \left< \,\,\overrightarrow{\mu}^{\prime}(0) \,\,\cdot \,\,\overrightarrow{\mu}^{\prime}(t) \right>\,,\tag{6}$$

where the autocorrelation function (i.e., �···�) is the thermal average of the scalar product of the dipole moment operator (i.e., −→*μ* ) of the dopant at two different times. This autocorrelation function can be resolved within the Liouville operator formalism to give [48]

$$\langle \begin{array}{c} \langle \ \overrightarrow{\mu}^{\dagger}(0) \ \cdot \ \overrightarrow{\mu}^{\dagger}(t) \ \rangle \equiv \exp\left[\rho\_{\mathbb{P}} \left\langle \mathsf{e}^{iL\_{\mathbb{S}}t} - 1 \right\rangle\_{\mathbb{S}} \right] \end{array} \rangle . \tag{7}$$

where the two-body Liouville operator *Lg* is defined by

$$L\_{\mathcal{S}}\Omega = H\_{\mathfrak{e}}\Omega - \Omega H\_{\mathfrak{g}} = \left[H\_{\mathfrak{g}\prime}\Omega\right] + \left(E\_{\mathfrak{e}} - E\_{\mathfrak{g}}\right)\Omega\,,\tag{8}$$

where Ω is an arbitrary operator. However, if lifetime broadening is neglected, only the dopant/perturber interaction and the dopant electronic energy change during the transition. Therefore, the autocorrelation function can be rewritten as [48]

$$\langle \langle \ \overrightarrow{\mu}^{\prime}(0) \cdot \ \overrightarrow{\mu}^{\prime}(t) \rangle \rangle \equiv \mathbf{e}^{i\omega\_{0}t} \exp\left[\rho\_{\mathrm{p}} \langle \mathbf{e}^{i\Delta Vt} - 1 \rangle\_{\mathcal{S}}\right] \tag{9}$$

where *ω*<sup>0</sup> is the transition frequency of the neat dopant, *ρ*<sup>P</sup> is the perturber density, and Δ*V* = *Ve* − *Vg*, with *Ve* and *Vg* being the excited-state dopant/ground-state perturber and ground-state dopant/ground-state perturber intermolecular potentials, respectively.

In semi-classical line shape theory, the line shape function for an allowed transition is given by [20],

$$\mathfrak{L}(\omega) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} dt \,\mathrm{e}^{-i\left[\omega(\mathbf{R}) - \omega\_0\right]t} \,\frac{Z\left(\beta\,V\_{\mathcal{S}} + i\,t\,\Delta V\right)}{Z\left(\beta\,V\_{\mathcal{S}}\right)}\,\mathrm{},\tag{10}$$

where *Z* is the partition function and **R** denotes the collection of all dopant/perturber distances. Under the classical fluid approximation of Percus [49–51], the autocorrelation function Φ(*t*) is given by a density expansion [20]

$$\Phi(t) = \ln Z\left(\beta V\_{\mathcal{S}} + it\,\Delta V\right) - \ln Z\left(\beta V\_{\mathcal{S}}\right) \\ = A\_1(t) + A\_2(t) + \cdots \\ \tag{11}$$

where [20, 25]

4 Will-be-set-by-IN-TECH

where *α* = *e*, *g* with *e* and *g* representing the excited state and ground state of the free dopant.

*N* ∑ *i*

where *N* is the total number of perturbing atoms in the range of the Rydberg optical electron.

[*Vα*(*ri*) + *V*�

perturber/perturber intermolecular potential. Therefore, the total Hamiltonian H is

which can be rewritten as a ground state Hamiltonian expectation value *Hg* and an excited

where the autocorrelation function (i.e., �···�) is the thermal average of the scalar product of the dipole moment operator (i.e., −→*μ* ) of the dopant at two different times. This autocorrelation

where Ω is an arbitrary operator. However, if lifetime broadening is neglected, only the dopant/perturber interaction and the dopant electronic energy change during the transition.

where *ω*<sup>0</sup> is the transition frequency of the neat dopant, *ρ*<sup>P</sup> is the perturber density, and Δ*V* = *Ve* − *Vg*, with *Ve* and *Vg* being the excited-state dopant/ground-state perturber and

In semi-classical line shape theory, the line shape function for an allowed transition is given

ground-state dopant/ground-state perturber intermolecular potentials, respectively.

*<sup>ρ</sup>*<sup>P</sup> �e*i Lg <sup>t</sup>* <sup>−</sup> <sup>1</sup>�*<sup>g</sup>*

*Lg* Ω = *He* Ω − Ω *Hg* = [*Hg*, Ω]+(*Ee* − *Eg*) Ω , (8)

*<sup>ρ</sup>*<sup>P</sup> �e*<sup>i</sup>* <sup>Δ</sup>*V t* <sup>−</sup> <sup>1</sup> �*<sup>g</sup>*

*dt* <sup>e</sup>−*<sup>i</sup>* [*ω*(**R**) <sup>−</sup> *<sup>ω</sup>*0] *<sup>t</sup> <sup>Z</sup>* (*<sup>β</sup> Vg* <sup>+</sup> *i t* <sup>Δ</sup>*V*)

*h*¯ 2 2 *m* ∇2

*<sup>i</sup>* , (3)

(*ri*)] |Ψ��Ψ| , (4)

*H* = *H*FD + *H*FP + *H*PD , (5)

*dt* <sup>e</sup>−*<sup>i</sup> <sup>ω</sup> <sup>t</sup>* � −→*<sup>μ</sup>* (0) · −→*<sup>μ</sup>* (*t*) � , (6)

, (7)

, (9)

*<sup>Z</sup>* (*<sup>β</sup> Vg*) , (10)

(*r*) is the

The Hamiltonian *H*FP for the free non-interacting perturber is given by [48]

Finally, the Hamiltonian *H*PD for the intermolecular interaction is [48]

*<sup>H</sup>*PD <sup>=</sup> <sup>∑</sup>*<sup>α</sup>* <sup>∑</sup>

state Hamiltonian expectation value *He*.

[48]

by [20],

*H*FP = −

*i*

The absorption coefficient function is given as the Fourier transform [48]

 ∞ −∞

function can be resolved within the Liouville operator formalism to give [48]

*μ* (*t*) � ≡ exp

2 *π*

<sup>L</sup>(*ω*) <sup>≡</sup> <sup>1</sup>

� −→

<sup>L</sup>(*ω*) = <sup>1</sup>

2*π*

 ∞ −∞

where the two-body Liouville operator *Lg* is defined by

*<sup>μ</sup>* (0) · −→

Therefore, the autocorrelation function can be rewritten as [48]

� −→*<sup>μ</sup>* (0) · −→*<sup>μ</sup>* (*t*) � ≡ <sup>e</sup>*<sup>i</sup> <sup>ω</sup>*<sup>0</sup> *<sup>t</sup>* exp

where *V*(*r*) is the dopant/perturber intermolecular potential, and *V*�

$$\begin{aligned} A\_{\boldsymbol{n}}(t) &= \frac{1}{n!} \int \cdots \int \prod\_{j=1}^{n} d^3 \mathbb{R}\_j \, \mathfrak{F}(\mathbf{R}\_1, \dots, \mathbf{R}\_n) \\ &\times \prod\_{j=1}^{n} \left[ \exp(-i \, \Delta V(\mathbf{R}\_j) \, t) - 1 \right] \quad . \end{aligned} \tag{12}$$

In eq. (12), F(**R**1,..., **R***n*) is the Ursell distribution function [25, 26, 49–51], and Δ*V*(**R**) = *Ve*(**R**) − *Vg*(**R**). The Ursell distribution function for two body interactions [25, 26] is F(**R**) = *ρ*<sup>P</sup> *g*PD(**R**), where *g*PD(**R**) is the perturber/dopant radial distribution function. The three body Ursell distribution function is estimated using the Kirkwood superposition approximation [52] to be

$$\mathfrak{F}(\mathbf{R}\_1, \mathbf{R}\_2) = \underset{\cdot}{\rho\_p^2} \left[ \mathfrak{g}\_{\mathbb{P}^\mathbb{P}}(|\mathbf{R}\_1 - \mathbf{R}\_2|) - 1 \right] \underset{\cdot}{\mathfrak{g}\_{\mathbb{P}^\mathbb{D}}(\mathbf{R}\_1)}{\operatorname{g}\_{\mathbb{P}^\mathbb{D}}(\mathbf{R}\_1)} \underset{\cdot}{\operatorname{g}\_{\mathbb{P}^\mathbb{D}}(\mathbf{R}\_2)} \tag{13}$$

where *g*PP(**R**) is the perturber/perturber radial distribution function. The density expansion terms are multibody interactions between dopant and perturber over all space. *A*1(*t*) is the dopant/perturber two-body interaction, *A*2(*t*) is the dopant/perturber three-body interaction, and *An*(*t*) is the dopant/perturber *n* + 1 body interaction. Substitution of these Ursell distribution functions into eq. (12) under the assumption of spherically symmetric potentials, gives [25, 26, 37]

$$A\_1(t) = 4\,\pi\rho\_{\mathbb{P}} \int\_0^\infty dr \, r^2 \, g\_{\mathbb{P}0}(r) \left[ \mathbf{e}^{-i t \Delta V(r)} - 1 \right] \,\tag{14}$$

and

$$\begin{split} A\_2(t) &= 4\pi \rho\_{\rm r}^2 \int\_0^\infty dr\_1 \, r\_1^2 \, g\_{\rm r0}(r\_1) \left[ \mathbf{e}^{-i t \Delta V(r\_1)} - 1 \right] \\ &\times \int\_0^\infty dr\_2 \, r\_2^2 \, g\_{\rm r0}(r\_2) \left[ \mathbf{e}^{-i t \Delta V(r\_2)} - 1 \right] \\ &\times \frac{1}{r\_1 \, r\_2} \int\_{|r\_1 - r\_2|}^{|r\_1 + r\_2|} \mathbf{s} \left[ \mathbf{g}\_{\rm r0}(s) - 1 \right] ds \,. \end{split} \tag{15}$$

Since the strength of the interaction decreases as the number of bodies involved increases, and since higher order interactions are more difficult to model, most line shape simulations are truncated at the second term *A*2(*t*).

The autocorrelation function can be written as a power series expansion at *t* = 0, namely [21, 48, 53, 54]

$$\Phi(t) \;= \sum\_{n=1}^{\infty} \frac{i^n}{n!} \; m\_n \; t^n \; \Big/ \tag{16}$$

where the expansion coefficients *mi* are given by

$$m\_{ll} = \left. \frac{1}{(\sqrt{-1})^n} \right|\_{dt^n} \frac{d^n}{dt^n} \Phi(t) \Big|\_{t=0} \,. \tag{17}$$

#### 6 Will-be-set-by-IN-TECH 456 Advanced Aspects of Spectroscopy

Using eq. (11) with *A*1(*t*) and *A*2(*t*) from eqs. (14) and (15), the first two expansion coefficients become [25]

$$m\_1 = -4\,\pi\rho\_\text{p} \int\_0^\infty dr \, r^2 \, g\_{\text{FD}}(r) \, \Delta V(r) \,\,\,\,\,\,\tag{18}$$

and

$$\begin{aligned} m\_2 &= m\_1^2 + 4 \,\pi \rho\_{\text{pr}} \int\_0^\infty dr \, r^2 \, g\_{\text{pr}}(r) \, \Delta V(r)^2 \\ &+ 8 \,\pi^2 \rho\_{\text{p}} \, ^2 \int\_0^\infty dr \, r\_1 \, dr\_2 \, r\_1 r\_2 \, g\_{\text{pr}}(r\_1) \, g\_{\text{pr}}(r\_2) \\ &\quad \times \, \Delta V(r\_1) \, \Delta V(r\_2) \int\_{|r\_1 - r\_2|}^{|r\_1 + r\_2|} \, s \, \left[ \, g\_{\text{pr}}(s) - 1 \right] ds \, . \end{aligned} \tag{19}$$

The above expansion coefficients are equivalent to the moments of the optical coefficient, which are defined as [25]

$$m\_{ll} = \int \mathfrak{L}(E) \to^n dE \,. \tag{20}$$

Thus, the perturber induced shift Δ(*ρ*P) in the energy position of the optical coefficient maximum is [25, 26]

$$
\Delta(\rho\_{\mathbb{P}}) = M\_1 = \frac{m\_1}{m\_0} = \int \mathfrak{L}(\mathbb{E}) \, \mathbb{E} \, d\mathbb{E} \, \Big/ \int \mathfrak{L}(\mathbb{E}) \, d\mathbb{E} \, \Big/ \tag{21}
$$

while the full-width-half-maximum of the experimental absorption spectrum is proportional to

$$M\_2 = \left[\frac{m\_2}{m\_0} - M\_1^2\right]^{1/2} = \left[\left(\int \mathfrak{L}(E) \, E^2 \, dE \, \bigg/ \int \mathfrak{L}(E) \, dE \,\right) - M\_1^2\right]^{1/2},\tag{22}$$

where L(*E*) is the absorption band for the transition and *E* = *h*¯(*ω* − *ω*0) [¯*h* ≡ reduced Planck constant].

#### **2.3. Previous studies**

The interaction of dopant low-*n* Rydberg states with dense fluids has previously been the subject of some interest both experimentally and theoretically [1, 22–36]. The first detailed investigation of these interactions was a study of Xe low-*n* Rydberg states doped into the dense rare gas fluids (i.e., argon, neon and helium) by Messing, *et al.* [25, 26]. In the same year, Messing, *et al.* [27, 28] presented their studies of molecular low-*n* Rydberg states in dense Ar and Kr. A decade later, Morikawa, *et al.* [34] probed the NO valence and low-*n* Rydberg state transitions in dense argon and krypton. All of these experiments [25–28, 34] used a basic moment analysis of absorption bands to determine the energy shifts of these bands as a function of perturber number density. The photoabsorption energy shifts were then simulated [25–28, 34] using eq. (18) for various assumptions of intermolecular potentials and radial distribution functions. Messing, *et al.* also performed line shape simulations under the assumption of a Gaussian line shape [25–28] for selected perturber number densities. As molecular dynamics developed, research groups [30, 31, 33, 35, 36] returned to absorption line shape simulations in an attempt to match the asymmetric broadening observed experimentally.

#### *2.3.1. Xe in Ar, Ne and He*

6 Will-be-set-by-IN-TECH

Using eq. (11) with *A*1(*t*) and *A*2(*t*) from eqs. (14) and (15), the first two expansion coefficients

*dr r*<sup>2</sup> *g*PD(*r*) Δ*V*(*r*)

 |*r*<sup>1</sup> + *r*2| |*r*<sup>1</sup> − *r*<sup>2</sup> |

The above expansion coefficients are equivalent to the moments of the optical coefficient,

Thus, the perturber induced shift Δ(*ρ*P) in the energy position of the optical coefficient

while the full-width-half-maximum of the experimental absorption spectrum is proportional

where L(*E*) is the absorption band for the transition and *E* = *h*¯(*ω* − *ω*0) [¯*h* ≡ reduced Planck

The interaction of dopant low-*n* Rydberg states with dense fluids has previously been the subject of some interest both experimentally and theoretically [1, 22–36]. The first detailed investigation of these interactions was a study of Xe low-*n* Rydberg states doped into the dense rare gas fluids (i.e., argon, neon and helium) by Messing, *et al.* [25, 26]. In the same year, Messing, *et al.* [27, 28] presented their studies of molecular low-*n* Rydberg states in dense Ar and Kr. A decade later, Morikawa, *et al.* [34] probed the NO valence and low-*n* Rydberg state transitions in dense argon and krypton. All of these experiments [25–28, 34] used a basic moment analysis of absorption bands to determine the energy shifts of these bands as a function of perturber number density. The photoabsorption energy shifts were then simulated [25–28, 34] using eq. (18) for various assumptions of intermolecular potentials and radial distribution functions. Messing, *et al.* also performed line shape simulations under the assumption of a Gaussian line shape [25–28] for selected perturber number densities. As molecular dynamics developed, research groups [30, 31, 33, 35, 36] returned to absorption line shape simulations in an attempt to match the asymmetric broadening

L(*E*) *E*<sup>2</sup> *dE*

L(*E*) *E dE*

2

*s* [ *g*PP(*s*) − 1 ] *ds* .

L(*E*) *E<sup>n</sup> dE* . (20)

<sup>−</sup> *<sup>M</sup>*<sup>2</sup> 1 1/2

L(*E*) *dE* , (21)

, (22)

*dr*<sup>1</sup> *dr*<sup>2</sup> *r*<sup>1</sup> *r*<sup>2</sup> *g*PD(*r*1) *g*PD(*r*2)

L(*E*) *dE*

*dr r*<sup>2</sup> *g*PD(*r*) Δ*V*(*r*) , (18)

(19)

 ∞ 0

 ∞ 0

> ∞ 0

*m*<sup>1</sup> = − 4 *π ρ*<sup>P</sup>

<sup>1</sup> + 4 *π ρ*<sup>P</sup>

2 ∞ 0

× Δ*V*(*r*1) Δ*V*(*r*2)

*mn* =

*m*<sup>0</sup> = 

= 

+ 8 *π*<sup>2</sup> *ρ*<sup>P</sup>

<sup>Δ</sup>(*ρ*P) = *<sup>M</sup>*<sup>1</sup> <sup>=</sup> *<sup>m</sup>*<sup>1</sup>

<sup>−</sup> *<sup>M</sup>*<sup>2</sup> 1 1/2

*m*<sup>2</sup> = *m*<sup>2</sup>

become [25]

which are defined as [25]

maximum is [25, 26]

*M*<sup>2</sup> =

**2.3. Previous studies**

observed experimentally.

 *m*<sup>2</sup> *m*<sup>0</sup>

to

constant].

and

**Figure 2.** Photoabsorption spectra (relative units) of the Xe 6*s* and 6*s*� transitions doped into Ar at an argon number density of *<sup>ρ</sup>*Ar <sup>=</sup> 1.47 <sup>×</sup> 1021 cm−<sup>3</sup> and a temperature of 23.6◦C. *<sup>a</sup>* corresponds to eq. (23), *<sup>b</sup>* to eq. (24), and *c* to eq. (25). For the 6*s* Rydberg state *E*<sup>0</sup> = 8.437 eV and for the 6*s*� Rydberg state *E*<sup>0</sup> = 9.570 eV. Data from the present work.

When Xe 6*s* and 6*s*� Rydberg state transitions are excited in the presence of low density argon or krypton, satellite bands appear on the higher energy wing of the absorption or emission spectra [22–24]. These blue satellite bands increase with increasing perturber density [22– 24] and with decreasing temperature. Therefore, Castex, *et al.* [22–24] concluded that these blue satellites are caused by the formation of dopant/perturber ground state and excited state dimers. An example of the Xe 6*s* and 6*s*� Rydberg transitions in the presence of argon is shown in Fig. 2. The primary Xe absorption transition, or [37]

$$\text{Xe} + h\nu \quad \xrightarrow{\text{Ar}} \quad \text{Xe}^\* \text{ \\_} \tag{23}$$

is indicated in Fig. 2 as *a*. The XeAr dimer transitions that yield the blue satellite bands are [22–24]

$$\text{\textasciicircum} + h\nu \quad \xrightarrow{\text{Ar}} \quad \text{\textasciicircum} + \text{Ar}\text{\textasciicircum}\tag{24}$$

and

$$\text{\color{red}{\text{XeAr}}} + \text{\color{red}{\text{hv}}} \quad \xrightarrow{\text{Ar}} \quad \text{\color{red}{\text{Xe}}^\*} \text{\color{red}{\text{Xe}}^\*} \text{Ar} \,. \tag{25}$$

These transitions are indicated in Fig. 2 as *b* and *c*, respectively.

Detailed investigations [25] of the Xe 6*s* Rydberg state doped in supercritical argon indicated that the energy position at the photoabsorption peak maximum shifted slightly to the red at low argon density and then strongly to the blue (cf. Fig. 3a). Similar results, which are shown in Fig. 3b, were then observed for the Xe 6*s*� Rydberg states in supercritical argon [26]. These studies [25, 26] concluded that both the perturber-induced energy shift and the line shape broadening were temperature independent. However, since the blue satellite bands grow and broaden as a function of perturber number density, the energy position of the maximum absorption (or the first moment from a moment analysis) is not an accurate energy position for the primary Xe Rydberg transition. Thus, modeling the experimental first moment *M*<sup>1</sup> and

**Figure 3.** The perturber induced shift Δ of (a) the Xe 6*s* [25] and (b) the Xe 6*s*� [26] absorption maximum plotted as a function of argon number density *ρ*Ar at different temperatures. In (a), the markers are () 25◦C; (◦) −83◦C; () −118◦C; () −138◦C; and (•) −163◦C. In (b), the markers are for temperature ranges of (×) −23◦C to 27◦C; (•) −93◦C to −63◦C; (◦) −123◦C to −113◦C; () −138◦C to −128◦C; () −158◦C to −148◦C; () −173◦C to −163◦C; and () −186◦C to −178◦C.

**Figure 4.** Photoabsorption spectra (relative units) of the CH3I 6*s* and 6*s*� transitions doped into Ar at an argon number density of *<sup>ρ</sup>*Ar <sup>=</sup> 1.89 <sup>×</sup> 1021 cm−<sup>3</sup> and a temperature of <sup>−</sup>79.8◦C. *<sup>a</sup>* corresponds to eq. (26) and *b* to eq. (27). For the 6*s* Rydberg state *E*<sup>0</sup> = 6.154 eV and *E*<sup>0</sup> = 6.767 eV for the 6*s*� Rydberg state. Data from the present work.

second moment *M*<sup>2</sup> using eqs. (19) and (20) required three groups of intermolecular potential parameters for different perturber density ranges.

The Xe 6*s* and 6*s*� Rydberg transitions [26] in supercritical helium and neon show a similar perturber density dependence as that shown in Fig. 3. However, these two systems do not form ground state or excited state dimers and, therefore, do not have blue satellite bands. Thus, the moment analysis of the photoabsorption spectra presented a more accurate perturber induced shift as a function of perturber number density. Because of the

**Figure 5.** The line shape simulation of the CH3I 6*s* transition doped into supercritical argon using (- - -) a semi-classical line shape function data [33] and using (···) molecular dynamics [32] in comparison to (—) the photoabsorption spectra at (a) *<sup>ρ</sup>*Ar <sup>=</sup> 2.0 <sup>×</sup> 1021 cm−3, (b) *<sup>ρ</sup>*Ar <sup>=</sup> 7.6 <sup>×</sup> 1021 cm−3, and (c) *<sup>ρ</sup>*Ar <sup>=</sup> 2.0 <sup>×</sup> 1022 cm−3. Experimental data are from the present work.

simplicity of the absorption bands, Messing, *et al.* [26] simulated the line shapes of the Xe 6*s* and 6*s*� transitions in both helium and neon at a single perturber number density using eqs. (10) and (11). These simulations indicated that an accurate line shape could be obtained without blue satellite bands [26]. Unfortunately, no temperature dependence was reported in these papers [25, 26].

#### *2.3.2. CH*3*I in Ar and Kr*

8 Will-be-set-by-IN-TECH

**Figure 3.** The perturber induced shift Δ of (a) the Xe 6*s* [25] and (b) the Xe 6*s*� [26] absorption maximum plotted as a function of argon number density *ρ*Ar at different temperatures. In (a), the markers are () 25◦C; (◦) −83◦C; () −118◦C; () −138◦C; and (•) −163◦C. In (b), the markers are for temperature ranges of (×) −23◦C to 27◦C; (•) −93◦C to −63◦C; (◦) −123◦C to −113◦C; () −138◦C to −128◦C; ()

**Figure 4.** Photoabsorption spectra (relative units) of the CH3I 6*s* and 6*s*� transitions doped into Ar at an argon number density of *<sup>ρ</sup>*Ar <sup>=</sup> 1.89 <sup>×</sup> 1021 cm−<sup>3</sup> and a temperature of <sup>−</sup>79.8◦C. *<sup>a</sup>* corresponds to eq. (26) and *b* to eq. (27). For the 6*s* Rydberg state *E*<sup>0</sup> = 6.154 eV and *E*<sup>0</sup> = 6.767 eV for the 6*s*� Rydberg state.

second moment *M*<sup>2</sup> using eqs. (19) and (20) required three groups of intermolecular potential

The Xe 6*s* and 6*s*� Rydberg transitions [26] in supercritical helium and neon show a similar perturber density dependence as that shown in Fig. 3. However, these two systems do not form ground state or excited state dimers and, therefore, do not have blue satellite bands. Thus, the moment analysis of the photoabsorption spectra presented a more accurate

perturber induced shift as a function of perturber number density. Because of the

−158◦C to −148◦C; () −173◦C to −163◦C; and () −186◦C to −178◦C.

Data from the present work.

parameters for different perturber density ranges.

Since CH3I is a molecular dopant, vibrational transitions as well as the adiabatic transition appear in the photoabsorption spectra. The adiabatic transition is given by [37]

$$\text{CH}\_3\text{I} + h\nu \quad \longrightarrow \quad \text{CH}\_3\text{I}^\*,\tag{26}$$

denoted *a* in Fig. 4, as well as one quantum of the CH3 deformation vibrational transition *ν*<sup>2</sup> in the excited state, or

$$\text{CH}\_3\text{I} + h\nu \quad \longrightarrow \quad \text{CH}\_3\text{I}^\*\left(\upsilon\_2\right), \tag{27}$$

denoted *b*, in Fig. 4.

Although vibrational transitions are apparent, CH3I [27–29, 32] has been investigated extensively because of the "atomic" like nature of the adiabatic 6*s* and 6*s*� Rydberg transitions. Messing, *et al.* [27, 28] extracted the perturber dependent shift Δ(*ρ*P) of the CH3I 6*s* and 6*s*� Rydberg states by performing a moment analysis on the photoabsorption bands using eq. (21). This analysis indicated that Δ(*ρ*P) tended first to lower energy and then to higher energy as *ρ*<sup>P</sup> increased from low density to the density of the triple-point liquid, similar to the trends shown in Fig. 3 for Xe in Ar. However, Messing, *et al.* [27, 28] did not explore critical temperature effects on Δ(*ρ*P), nor did they correctly account for the vibrational bands on the blue side of the adiabatic Rydberg transition. Messing, *et al.* [28] did attempt to model the CH3I 6*s* Rydberg transition in argon using a semi-classical statistical line shape function under the assumption that the adiabatic and vibrational transitions have exactly the same line shapes, although no comparison between the experimental spectra and the simulated line shapes was provided. Later researchers [30–33] concentrated on the simulation of the CH3I 6*s* Rydberg state doped into argon using both molecular dynamics and semi-classical integral methods. Egorov, *et al.* [33] showed that the semi-classical integral method can yield results comparable to the molecular dynamics calculations of Ziegler, *et al.* [30–32]. Comparing the semi-classical method and molecular dynamics simulation to experimental spectra (cf. Fig. 5) shows that

**Figure 6.** Experimental (•) and calculated (—) energy shift of the NO (a) *<sup>B</sup>*�2<sup>Δ</sup> <sup>←</sup> *<sup>X</sup>*2<sup>Π</sup> (*v*� = 7, 0) valence transition, (b) *<sup>A</sup>*2Σ<sup>+</sup> <sup>←</sup> *<sup>X</sup>*2<sup>Π</sup> (*v*� = 0, 0) Rydberg transition and (c) *<sup>C</sup>*2Π<sup>+</sup> <sup>←</sup> *<sup>X</sup>*2<sup>Π</sup> (*v*� = 0, 0) Rydberg transition plotted versus the reduced argon number density [34].

both methods can be used to simulate the experimental spectra with an appropriate choice of intermolecular potentials.

#### *2.3.3. NO in Ar*

Morikawa, *et al.* [34] investigated valence and low-*n* Rydberg transitions doped into supercritical argon. They used eq. (21) to determine the perturber induced shift Δ(*ρ*P) for several low-*n* Rydberg transitions as well as a valence transition. Under the assumption of spherically symmetric potentials for the ground and excited states of the NO/Ar systems, an accurate moment analysis using eq. (18) was performed (cf. Fig. 6). In The NO valence state transition (cf. Fig. 6a) shows a slight perturber-induced red shift, which differs significantly from the obvious blue shift of the low-*n* NO Rydberg state transitions shown in Figs. 6b and c. Fig. 6 also shows that low-*n* Rydberg states make a more sensitive probe to perturber effects than valence transitions. Later groups [35, 36] did line shape simulations to model the experimental spectra. Larrégaray, *et al.* [35] measured the NO 3*sσ* transition doped into supercritical argon at selected argon densities. Then, using molecular dynamics they successfully modeled the line shape of the transition. Once the intermolecular potentials and boundary conditions for the molecular dynamics simulations had been set against experimental data, Larrégaray, *et al.* [35] calculated the line shape for NO in Ar at the critical density and temperature. These calculations predicted that the photoabsorption peak maximum position would shift to the blue when argon was near its critical point. Later, Egorov, *et al.* [36] showed that similar results could be obtained using the semi-classical approximation. Therefore, line shape simulations using molecular dynamics and semi-classical line shape theory show comparable results.

#### **3. Experiment techniques and theoretical methodology**

#### **3.1. Experimental techniques**

All of the photoabsorption measurements presented in Sections 4 and 5 were obtained using the one-meter aluminum Seya-Namioka (Al-SEYA) beamline on bending magnet 8 of the Aladdin storage ring at the University of Wisconsin Synchrotron Radiation Center (SRC) in Stoughton, WI. This beamline, which was decommissioned during Winter 2007, produced monochromatic synchrotron radiation having a resolution of ∼ 8 meV in the energy range of 6 - 11 eV. The monochromatic synchrotron radiation enters the vacuum chamber which is equipped with a sample cell (cf. Fig. 7). The photoabsorption signal is detected by a photomultiplier that is connected to the data collection computer via a Keithley 486 picoammeter. The pressure in the vacuum chamber is maintained at low 10−<sup>8</sup> to high 10−<sup>9</sup>

**Figure 7.** Schematic of the copper experimental cell.

10 Will-be-set-by-IN-TECH

**Figure 6.** Experimental (•) and calculated (—) energy shift of the NO (a) *<sup>B</sup>*�2<sup>Δ</sup> <sup>←</sup> *<sup>X</sup>*2<sup>Π</sup> (*v*� = 7, 0) valence transition, (b) *<sup>A</sup>*2Σ<sup>+</sup> <sup>←</sup> *<sup>X</sup>*2<sup>Π</sup> (*v*� = 0, 0) Rydberg transition and (c) *<sup>C</sup>*2Π<sup>+</sup> <sup>←</sup> *<sup>X</sup>*2<sup>Π</sup> (*v*� = 0, 0) Rydberg

both methods can be used to simulate the experimental spectra with an appropriate choice of

Morikawa, *et al.* [34] investigated valence and low-*n* Rydberg transitions doped into supercritical argon. They used eq. (21) to determine the perturber induced shift Δ(*ρ*P) for several low-*n* Rydberg transitions as well as a valence transition. Under the assumption of spherically symmetric potentials for the ground and excited states of the NO/Ar systems, an accurate moment analysis using eq. (18) was performed (cf. Fig. 6). In The NO valence state transition (cf. Fig. 6a) shows a slight perturber-induced red shift, which differs significantly from the obvious blue shift of the low-*n* NO Rydberg state transitions shown in Figs. 6b and c. Fig. 6 also shows that low-*n* Rydberg states make a more sensitive probe to perturber effects than valence transitions. Later groups [35, 36] did line shape simulations to model the experimental spectra. Larrégaray, *et al.* [35] measured the NO 3*sσ* transition doped into supercritical argon at selected argon densities. Then, using molecular dynamics they successfully modeled the line shape of the transition. Once the intermolecular potentials and boundary conditions for the molecular dynamics simulations had been set against experimental data, Larrégaray, *et al.* [35] calculated the line shape for NO in Ar at the critical density and temperature. These calculations predicted that the photoabsorption peak maximum position would shift to the blue when argon was near its critical point. Later, Egorov, *et al.* [36] showed that similar results could be obtained using the semi-classical approximation. Therefore, line shape simulations using molecular dynamics

All of the photoabsorption measurements presented in Sections 4 and 5 were obtained using the one-meter aluminum Seya-Namioka (Al-SEYA) beamline on bending magnet 8 of the Aladdin storage ring at the University of Wisconsin Synchrotron Radiation Center (SRC) in Stoughton, WI. This beamline, which was decommissioned during Winter 2007, produced monochromatic synchrotron radiation having a resolution of ∼ 8 meV in the energy range of 6 - 11 eV. The monochromatic synchrotron radiation enters the vacuum chamber which is equipped with a sample cell (cf. Fig. 7). The photoabsorption signal is detected by a photomultiplier that is connected to the data collection computer via a Keithley 486 picoammeter. The pressure in the vacuum chamber is maintained at low 10−<sup>8</sup> to high 10−<sup>9</sup>

transition plotted versus the reduced argon number density [34].

and semi-classical line shape theory show comparable results.

**3.1. Experimental techniques**

**3. Experiment techniques and theoretical methodology**

intermolecular potentials.

*2.3.3. NO in Ar*

Torr by a Perkin-Elmer ion pump. The experimental copper cell is equipped with entrance and exit MgF2 windows (with an energy cut-off of 10.9 eV) that are capable of withstanding pressures of up to 100 bar and temperatures of up to 85◦C. This cell, which has a path length of 1.0 cm, is connected to an open flow liquid nitrogen cryostat and resistive heater system allowing the temperature to be controlled to within ± 0.5◦C with a Lakeshore 330 Autotuning Temperature controller. The gas sample is added through a gas handling system (GHS), which consists of 316-stainless steel components connected by copper gasket sealed flanges. The initial pressure for the GHS and sample cell is in the low 10−<sup>8</sup> Torr range.

During the initial bakeout, the GHS and the vacuum chamber were heated to 100◦C under vacuum for several days to remove any water adsorbed onto the surface of the stainless steel. The initial bake was stopped when the base pressure of the GHS and sample chamber is in the low 10−<sup>7</sup> or high 10−<sup>8</sup> Torr range, so that upon cooling the final GHS base pressure was <sup>10</sup>−<sup>8</sup> <sup>−</sup> <sup>10</sup>−<sup>9</sup> Torr. Anytime a system was changed (either the dopant or the perturber), the GHS was again baked in order to return the system to near the starting base pressure. This prevented cross-contamination between dopant/perturber systems.

The intensity of the synchrotron radiation exiting the monochromator was monitored by recording the beam current of the storage ring as well as the photoemission from a nickel mesh situated prior to the sample cell. The light then entered the experimental cell through a MgF2 window, traveled through the sample and then a second MgF2 window (cf. Fig. 7) before striking a thin layer of sodium salicylate powder on the inside of a glass window that preceded the photomultiplier tube. An empty cell (acquired for a base pressure of 10−<sup>7</sup> or 10−<sup>8</sup> Torr) spectrum was used to correct the dopant absorption spectra for the monochromator flux, for absorption by the MgF2 windows, and for any fluctuations in the quantum efficiency of the sodium salicylate window.

Two dopants (i.e. methyl iodide and xenon) and five perturbers (i.e. argon, krypton, xenon, carbon tetrafluoride and methane) were investigated. All dopants and perturbers were used without further purification: methyl iodide (Aldrich, 99.45%), argon (Matheson Gas Products, 99.9999%), krypton (Matheson Gas Products, 99.998%), and xenon (Matheson Gas Products, 99.995%). When CH3I was the dopant, the CH3I was degassed with three freeze/pump/thaw cycles prior to use. Photoabsorption spectra for each neat dopant and each neat perturber were measured to verify the absence of impurities in the spectral range of interest. The atomic perturber number densities were calculated from the Strobridge equation of state [55] with the parameters obtained from [56] for argon, [57] for krypton, and [58] for xenon. All densities and temperatures were selected to maintain a single phase system in the sample cell. At temperatures below *Tc*, a change in density required a change in temperature, since the isotherms are steeply sloped.

The sensitivity of the absorption spectra to local density required that the quality of a data set be monitored by performing basic data analysis during measurements. Any anomalies were corrected by immediately re-measuring the photoabsorption spectrum for the problem density/temperature/pressure after allowing additional time for increased density stabilization. Once a data set was obtained for non-critical temperatures, the photoabsorption data for perturber densities on an isotherm near the critical isotherm were then measured. For the near critical data set we selected a temperature that was +0.5◦C above the critical temperature (chosen to prevent liquid formation in the cell during temperature stabilization near the critical density and to minimize critical opalescence during data acquisition). Near the critical density, the consistency of the density step is dependent on the slope of the critical isotherm. If the critical isotherm has a small slope in this region, it becomes difficult to acquire samples at a constant density step size due to our inability to vary the perturber pressure practically by less than 0.01 bar and to the difficulties encountered in maintaining temperature stability. For instance, near the critical density of xenon, a 1 mbar change in pressure or a 0.001◦C change in temperature causes a density change of 2.0 <sup>×</sup> 1021 cm−3. Maintaining the necessary temperature stability (i.e., ± 0.2◦C) during the acquisition of data along the critical isotherm is difficult with an open flow liquid nitrogen cryostat system and usually required constant monitoring with manual adjustment of the nitrogen flow.

#### **3.2. Theoretical methodology**

#### *3.2.1. Line shape function*

The experimental line shapes were simulated using the semi-classical statistical line shape function given in eq. (10). Rewriting eq. (10) in terms of the autocorrelation function allows eq. (10) to be given as a Fourier transform, namely [25, 26, 33, 37]

$$\mathfrak{L}(\omega) = \frac{1}{2\pi} \operatorname{Re} \int\_{-\infty}^{\infty} dt \,\mathrm{e}^{-i\omega \cdot t} \left< \mathrm{e}^{i\omega(\mathbf{R}) \cdot t} \right>\,,\tag{28}$$

where *ω* = *ω*(**R**) − *ω*0, with *ω*<sup>0</sup> being the transition frequency for the neat dopant. Eq. (28) neglects lifetime broadening and assumes that the transition dipole moment is independent of **R**. In the substitution of the exponential density expansion [i.e., eq. (11)] for the autocorrelation function, the general term *An* represents a (n+1)-body interaction [20]. However, since the strength of the interaction decreases as the number of bodies involved increases, and since the higher order interactions are more difficult to model, our line shape simulations are truncated at the second term *A*2(*t*), or three body interactions. Within this approximation, eq. (28) becomes

$$\mathfrak{L}(\omega) = \frac{1}{2\pi} \operatorname{Re} \int\_{-\infty}^{\infty} dt \,\mathrm{e}^{-i\omega t} \,\exp\left[A\_1(t) + A\_2(t)\right] \tag{29}$$

where the two terms are recalled from eq. (14),

$$A\_1(t) = 4\pi \rho\_{\rm p} \int\_0^\infty dr \, r^2 \, g\_{\rm FD}(r) \left[ \mathbf{e}^{-i t \Delta V(r)} - 1 \right] \quad \text{(7)}$$

and eq. (15),

12 Will-be-set-by-IN-TECH

At temperatures below *Tc*, a change in density required a change in temperature, since the

The sensitivity of the absorption spectra to local density required that the quality of a data set be monitored by performing basic data analysis during measurements. Any anomalies were corrected by immediately re-measuring the photoabsorption spectrum for the problem density/temperature/pressure after allowing additional time for increased density stabilization. Once a data set was obtained for non-critical temperatures, the photoabsorption data for perturber densities on an isotherm near the critical isotherm were then measured. For the near critical data set we selected a temperature that was +0.5◦C above the critical temperature (chosen to prevent liquid formation in the cell during temperature stabilization near the critical density and to minimize critical opalescence during data acquisition). Near the critical density, the consistency of the density step is dependent on the slope of the critical isotherm. If the critical isotherm has a small slope in this region, it becomes difficult to acquire samples at a constant density step size due to our inability to vary the perturber pressure practically by less than 0.01 bar and to the difficulties encountered in maintaining temperature stability. For instance, near the critical density of xenon, a 1 mbar change in pressure or a 0.001◦C change in temperature causes a density change of 2.0 <sup>×</sup> 1021 cm−3. Maintaining the necessary temperature stability (i.e., ± 0.2◦C) during the acquisition of data along the critical isotherm is difficult with an open flow liquid nitrogen cryostat system and usually required

The experimental line shapes were simulated using the semi-classical statistical line shape function given in eq. (10). Rewriting eq. (10) in terms of the autocorrelation function allows

where *ω* = *ω*(**R**) − *ω*0, with *ω*<sup>0</sup> being the transition frequency for the neat dopant. Eq. (28) neglects lifetime broadening and assumes that the transition dipole moment is independent of **R**. In the substitution of the exponential density expansion [i.e., eq. (11)] for the autocorrelation function, the general term *An* represents a (n+1)-body interaction [20]. However, since the strength of the interaction decreases as the number of bodies involved increases, and since the higher order interactions are more difficult to model, our line shape simulations are truncated at the second term *A*2(*t*), or three body interactions. Within this

*dt* <sup>e</sup>−*<sup>i</sup> <sup>ω</sup> <sup>t</sup>* �e*<sup>i</sup> <sup>ω</sup>*(**R**)*<sup>t</sup>* � , (28)

*dt* e<sup>−</sup> *<sup>i</sup> <sup>ω</sup> <sup>t</sup>* exp [*A*1(*t*) + *A*2(*t*)] , (29)

 ,

constant monitoring with manual adjustment of the nitrogen flow.

eq. (10) to be given as a Fourier transform, namely [25, 26, 33, 37]

<sup>L</sup>(*ω*) = <sup>1</sup>

2 *π* Re ∞ −∞

 ∞ 0

*dr r*<sup>2</sup> *g*PD(*r*)

<sup>e</sup><sup>−</sup> *i t* <sup>Δ</sup>*V*(*r*) <sup>−</sup> <sup>1</sup>

isotherms are steeply sloped.

**3.2. Theoretical methodology**

approximation, eq. (28) becomes

<sup>L</sup>(*ω*) = <sup>1</sup>

where the two terms are recalled from eq. (14),

2*π* Re ∞ −∞

*A*1(*t*) = 4 *π ρ*<sup>P</sup>

*3.2.1. Line shape function*

$$\begin{split} A\_{2}(t) &= 4\pi \rho\_{\mathrm{p}}^{2} \int\_{0}^{\infty} dr\_{1} \, r\_{1}^{2} \, g\_{\mathrm{pr}}(r\_{1}) \left[ \mathbf{e}^{-it\,\Delta V(r\_{1})} - 1 \right] \\ &\times \int\_{0}^{\infty} dr\_{2} \, r\_{2}^{2} \, g\_{\mathrm{pr}}(r\_{2}) \left[ \mathbf{e}^{-it\,\Delta V(r\_{2})} - 1 \right] \\ &\times \frac{1}{r\_{1}r\_{2}} \int\_{|r\_{1}-r\_{2}|}^{|r\_{1}+r\_{2}|} s \left[ \, g\_{\mathrm{pr}}(s) - 1 \right] ds \,. \end{split}$$

The required radial distribution functions (i.c., *g*PP and *g*PD) were obtained from the analytical solution of the Ornstein-Zernike equation for a binary system within the Percus-Yevick (PY) closure [59], while the Fourier transform for eq. (29) was performed using a standard fast Fourier transform algorithm [60]. The line shape obtained from the transform of eq. (29) was convoluted with a standard Gaussian slit function to account for the finite resolution (∼ 8 meV) of the monochromator. More detailed discussions are given below.

#### *3.2.2. Fast Fourier transform*

A Fourier transform has the general form [60]

$$F(\omega) = \frac{1}{2\,\pi} \int\_{-\infty}^{\infty} f(t) \,\mathrm{e}^{-i\omega t} \, dt \,\,. \tag{30}$$

Since the line shape function is calculated numerically, the integration limits for eq. (30) must be finite and, therefore, an appropriate integration range must be determined. For any Fourier transformation, the integration limit and the total number of steps are related through [60]

$$
\delta t \times \delta \omega = \frac{2\pi}{N} \,\prime \tag{31}
$$

where *δ* stands for the sampling interval (i.e., the step size) of the corresponding variable and *N* is the total number of discrete points. Fourier transforms rely on the fact that data are usually obtained in discrete steps and the generating functions *f*(*t*) and *F*(*ω*) can be represented by the set of points

$$\begin{aligned} f\_k &\equiv f(t\_k), \quad t\_k = k\delta t\_\prime \quad k = 1, \dots, N, \\ F\_\hbar &\equiv F(\omega\_\hbar), \quad \omega\_\hbar = \hbar \delta \omega\_\prime \quad \hbar = -\frac{N}{2}, \dots, \frac{N}{2} - 1 \end{aligned} \tag{32}$$

Therefore, the function *F*(*ω*) is determined point-wise using

$$\begin{split} F(\omega\_{\boldsymbol{n}}) &= \frac{1}{2\,\pi} \int\_{-\infty}^{\infty} f(t) \, \mathbf{e}^{-i\omega\_{\boldsymbol{n}}t} \, dt = \frac{1}{2\,\pi} \sum\_{k=1}^{N} f\_{k} \, \mathbf{e}^{-i\omega\_{\boldsymbol{n}}t\_{k}} \, \delta t \\ &= \frac{\delta t}{2\,\pi} \sum\_{k=1}^{N} f\_{k} \, \mathbf{e}^{-i2\,\pi \,\boldsymbol{n}k/N} \, . \end{split} \tag{33}$$

For simplicity, we will define the discrete Fourier transform from time to angular frequency as eq. (33). When computing the Fourier transform from eq. (33), the quickest method is known as a fast Fourier transform (FFT) and requires that the number of steps *N* be a power of 2. In our calculations, we use a Cooley-Tukey FFT algorithm [60] with *N* = 1024. The requirements for calculating eq. (30) within this FFT algorithm are, therefore, a complex array of the calculated values of the time dependent autocorrelation function truncated to the second term.

Rewriting eq. (29) using Euler's relation yields [38]

$$
\langle \mathbf{e}^{i\omega(\mathbf{R})t} \rangle = \operatorname{Re} \langle \mathbf{e}^{i\omega(\mathbf{R})t} \rangle + \operatorname{Im} \langle \mathbf{e}^{i\omega(\mathbf{R})t} \rangle \,\tag{34}
$$

where the real and the imaginary parts are given by

$$\begin{aligned} \left| \text{Re} \left\langle e^{i\omega(\mathbf{R})t} \right\rangle \right| &= \exp \left[ \text{Re} \left( A\_1(t) + A\_2(t) \right) \right] \cos \left[ \text{Im} \left( A\_1(t) + A\_2(t) \right) \right] \\\\ \left| \text{Im} \left\langle e^{i\omega(\mathbf{R})t} \right\rangle \right| &= \exp \left[ \text{Re} \left( A\_1(t) + A\_2(t) \right) \right] \sin \left[ \text{Im} \left( A\_1(t) + A\_2(t) \right) \right] \, . \end{aligned} \tag{35}$$

In eq. (35),

$$\begin{aligned} \text{Re}[A\_1(t) + A\_2(t)] \\ &= 4\pi \rho\_r \int\_0^\infty dr \, r^2 \, g\_{\text{FD}}(r) \left[ \cos(\Delta V(r) \, t) - 1 \right] \\ &+ 4\pi \rho\_r^2 \int\_0^\infty dr\_1 \, dr\_2 \, h(r\_1, r\_2) \left[ \cos(\Delta V(r\_1) \, t) \, \cos(\Delta V(r\_2) \, t) \\ &\quad + 1 - \sin(\Delta V(r\_1) \, t) \, \sin(\Delta V(r\_2) \, t) \\ &\quad - \cos(\Delta V(r\_1) \, t) \, - \cos(\Delta V(r\_2) \, t) \right], \end{aligned} \tag{36}$$

and

$$\begin{aligned} \text{Im}[A\_1(t) + A\_2(t)] \\ &= -4 \,\pi \rho\_{\text{F}} \int\_0^\infty dr \, r^2 \, g\_{\text{FD}}(r) \, \left[ \sin(\Delta V(r) \, t) \right] \\ &- 4 \,\pi \rho\_{\text{F}}^2 \int\_0^\infty dr\_1 \, dr\_2 \, h(r\_1, r\_2) \left[ \sin(\Delta V(r\_1) \, t) \, \cos(\Delta V(r\_2) \, t) \right. \\ &\left. + \cos(\Delta V(r\_1) \, t) \, \sin(\Delta V(r\_2) \, t) \right. \\ &\left. - \sin(\Delta V(r\_1) \, t) \, - \sin(\Delta V(r\_2) \, t) \right] \, , \end{aligned} \tag{37}$$

with

$$h(r\_1, r\_2) = r\_1 \ g\_{\text{FD}}(r\_1) \ r\_2 \ g\_{\text{FD}}(r\_2) \ \int\_{|r\_1 - r\_2|}^{|r\_1 + r\_2|} s \begin{bmatrix} \mathbf{g}\_{\text{FF}}(\mathbf{s}) \ -\mathbf{1} \end{bmatrix} \, ds \ .$$

The output of the FFT is a complex function of frequency. The real portion of this complex function is obtained and then convoluted with a standard Gaussian slit function. The final output is the simulated line shape function. Since eqs. (29), (36) and (37) depend on both the radial distribution functions and the ground-state and excited-state intermolecular potentials, these are discussed in more detail below.

#### *3.2.3. Radial distribution function*

After significant investigation, we found that the most stable calculation technique for obtaining radial distribution functions for this problem was the analytical solution of the Ornstein-Zernike relation within the Percus-Yevick (PY) closure [59]. Although this solution for a binary system yields four coupled integro-differential equations, dilute solutions (i.e., *ρ*<sup>D</sup> � *ρ*P) allows these equations to be reduced to the calculation of the perturber/dopant radial distribution function *g*PD(*r*) and the perturber/perturber radial distribution function *g*PP(*r*). This solution is given by [12, 59]

$$\begin{cases} \mathbf{g\_{FD}}(r) = \left. r^{-1} \mathbf{e^{-\beta V\_{\mathcal{S}}(r)}} \mathbf{y\_{FD}}(r) \right| \\\\ \mathbf{g\_{\mathcal{FP}}}(r) = \left. r^{-1} \mathbf{e^{-\beta V\_{\mathcal{S}}'(r)}} \mathbf{y\_{\mathcal{FP}}}(r) \right| \end{cases} \tag{38}$$

where

(35)

(36)

(37)

14 Will-be-set-by-IN-TECH

known as a fast Fourier transform (FFT) and requires that the number of steps *N* be a power of 2. In our calculations, we use a Cooley-Tukey FFT algorithm [60] with *N* = 1024. The requirements for calculating eq. (30) within this FFT algorithm are, therefore, a complex array of the calculated values of the time dependent autocorrelation function truncated to

�e*<sup>i</sup> <sup>ω</sup>*(**R**)*<sup>t</sup>* � <sup>=</sup> Re �e*<sup>i</sup> <sup>ω</sup>*(**R**)*<sup>t</sup>* � <sup>+</sup> Im �e*<sup>i</sup> <sup>ω</sup>*(**R**)*<sup>t</sup>* � , (34)

*dr*<sup>1</sup> *dr*<sup>2</sup> *h*(*r*1,*r*2) [cos(Δ*V*(*r*1) *t*) cos(Δ*V*(*r*2) *t*)

+ 1 − sin(Δ*V*(*r*1) *t*) sin(Δ*V*(*r*2) *t*)

*dr*<sup>1</sup> *dr*<sup>2</sup> *h*(*r*1,*r*2) [sin(Δ*V*(*r*1) *t*) cos(Δ*V*(*r*2) *t*)

+ cos(Δ*V*(*r*1) *t*) sin(Δ*V*(*r*2) *t*)

 |*r*<sup>1</sup> + *r*2| |*r*<sup>1</sup> − *r*2|

− cos(Δ*V*(*r*1) *t*) − cos(Δ*V*(*r*2) *t*)] ,

− sin(Δ*V*(*r*1) *t*) − sin(Δ*V*(*r*2) *t*)] ,

*s*[*g*PP(*s*) − 1 ] *ds* .

� = exp [Re (*A*1(*t*) + *A*2(*t*)) ] cos [Im (*A*1(*t*) + *A*2(*t*)) ] ,

� = exp [Re (*A*1(*t*) + *A*2(*t*)) ] sin [Im (*A*1(*t*) + *A*2(*t*)) ] .

*dr r*<sup>2</sup> *<sup>g</sup>*PD(*r*) [cos(Δ*V*(*r*) *<sup>t</sup>*) <sup>−</sup> <sup>1</sup> ]

*dr r*<sup>2</sup> *g*PD(*r*) [sin(Δ*V*(*r*) *t*)]

The output of the FFT is a complex function of frequency. The real portion of this complex function is obtained and then convoluted with a standard Gaussian slit function. The final output is the simulated line shape function. Since eqs. (29), (36) and (37) depend on both the radial distribution functions and the ground-state and excited-state intermolecular potentials,

the second term.

In eq. (35),

and

with

Rewriting eq. (29) using Euler's relation yields [38]

where the real and the imaginary parts are given by

Re �*e<sup>i</sup> <sup>ω</sup>*(**R**)*<sup>t</sup>*

Im �*e<sup>i</sup> <sup>ω</sup>*(**R**)*<sup>t</sup>*

Re[*A*1(*t*) + *A*2(*t*) ]

= 4 *π ρ*<sup>P</sup>

+ 4*π ρ*<sup>2</sup> P ∞ 0

Im[*A*1(*t*) + *A*2(*t*) ]

= − 4*π ρ*<sup>P</sup>

<sup>−</sup> <sup>4</sup> *π ρ*<sup>2</sup> P ∞ 0

these are discussed in more detail below.

 ∞ 0

> ∞ 0

 ∞ 0

> ∞ 0

*h*(*r*1,*r*2) = *r*<sup>1</sup> *g*PD(*r*1) *r*<sup>2</sup> *g*PD(*r*2)

$$\begin{aligned} \left(Y\_{\rm FD}(r)\right) &= \int\_0^r dt \, \frac{dY\_{\rm FD}(t)}{dt} , \\\\ \left(Y\_{\rm FD}(r)\right) &= \int\_0^r dt \, \frac{dY\_{\rm FD}(t)}{dt} , \end{aligned} \tag{39}$$

with

$$\begin{aligned} \frac{d}{dr} Y\_{\rm pr}(r) &= 1 + 2\pi \rho\_{\rm pr} \int\_0^\infty dt \left( \mathbf{e}^{-\beta V\_{\rm \tilde{b}}(t)} - 1 \right) Y\_{\rm pr}(t) \\\\ &\times \left[ \mathbf{e}^{-\beta V\_{\rm \tilde{b}}'(r+t)} \mathbf{y}\_{\rm pr}(r+t) \right. \\\\ &\left. - \frac{r-t}{|r-t|} \mathbf{e}^{-\beta V\_{\rm \tilde{b}}'(|r-t|)} \mathbf{y}\_{\rm pr}(|r-t|) - 2t \right], \\\\ \frac{d}{dr} Y\_{\rm pr}(r) &= 1 + 2\pi \rho\_{\rm r} \int\_0^\infty dt \left( \mathbf{e}^{-\beta V\_{\rm \tilde{b}}'(t)} - 1 \right) Y\_{\rm pr}(t) \end{aligned} \tag{40}$$

$$\begin{aligned} \mathcal{Y} &= \mathcal{Y} \\\\ \times \left[ \mathbf{e}^{-\beta' V\_{\text{gr}}'(r+t)} \, \mathcal{Y}\_{\text{pr}}(r+t) \\\\ &- \frac{r-t}{|r-t|} \, \mathbf{e}^{-\beta' V\_{\text{g}}'(|r-t|)} \, \mathcal{Y}\_{\text{pr}}(|r-t|) - 2t \, t \right], \end{aligned}$$

and with *V*g and *V*� <sup>g</sup> being the ground state perturber/dopant and ground state perturber/perturber intermolecular potentials, respectively.

#### *3.2.4. Intermolecular potentials*

Eqs. (29), (34) - (40) are explicitly dependent on the excited-state and ground-state perturber/dopant intermolecular potentials through Δ*V*(*r*), and are implicitly dependent on the perturber/perturber and perturber/dopant ground-state intermolecular potential via *g*PP(*r*) and *g*PD(*r*). Thus, these simulations require one to develop a single set of ground-state and excited-state intermolecular potential parameters for each system. A standard Lennard-Jones 6-12 potential, or

$$V(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right],\tag{41}$$

was chosen for the atomic perturber/perturber ground-state intermolecular interactions, and non-polar dopant/perturber ground-state intermolecular interactions. The ground-state molecular perturber/perturber intermolecular potential was a two-Yukawa potential, or

$$V(r) = -\frac{\kappa\_0 \varepsilon}{r} \left[ \mathbf{e}^{-z\_1 \left( r - \sigma \right)} - \mathbf{e}^{-z\_2 \left( r - \sigma \right)} \right] \tag{42}$$

The ground-state polar dopant/perturber intermolecular interactions were modeled using a modified Stockmeyer potential

$$V(r) = 4\epsilon' \left[ \left(\frac{\sigma'}{r}\right)^{12} - \left(\frac{\sigma'}{r}\right)^6 \right] - \frac{1}{r^6} a\_p \mu\_D^2 \tag{43}$$

which can be rewritten in standard Lennard-Jones 6-12 potential form [12], with

$$
\varepsilon = \varepsilon' \left[ 1 + \frac{\alpha\_p \,\mu\_D^{\,2}}{4 \varepsilon' \sigma'^6} \right]^2
$$
 
$$
\sigma = \sigma' \left[ 1 + \frac{\alpha\_p \,\mu\_D^{\,2}}{4 \varepsilon' \sigma'^6} \right]^{-1/6}
$$

(The modified Stockmeyer potential includes orientational effects via an angle average that presumes the free rotation of the polar dopant molecule.) An exponential-6 potential, given by

$$V(r) = \frac{\varepsilon}{1 - (6/\gamma)} \left\{ \frac{6}{\gamma} e^{\gamma(1-\chi)} - \chi^{-6} \right\} \,\,\,\,\tag{44}$$

.

was chosen for the excited-state dopant/ground-state perturber interactions. In eqs. (41) - (44), *ε* is the well depth, *σ* is the collision parameter, *α<sup>P</sup>* is the perturber polarizability, *μ<sup>D</sup>* is dopant dipole moment, *χ* ≡ *r*/*re* (where *re* is the equilibrium distance), and *γ* is the potential steepness.

The Lennard-Jones parameters for the atomic fluids and the modified Stockmeyer parameters for the CH3I/Ar and CH3I/Kr interactions were identical to the parameters used to model accurately the perturber-induced shift of the dopant ionization energy for methyl iodide in argon [12–14, 61], krypton [12, 15, 61] and xenon [16, 61]. The parameters *κ*0, *z*1, *z*2, *ε* and *σ* in eq. (42) were adjusted to give the best fit to the phase diagram of the perturber [61].The parameters *ε*, *σ*, *χ* and *γ* in eq. (44) were adjusted by hand to give the "best" fit to the experimental absorption spectra of the dopant low-*n* Rydberg states in each of the fluids investigated here.

**Figure 8.** Selected photoabsorption spectra (–, relative scale) and simulated line shapes (··· ) for the Xe 6*s* Rydberg transitions at (a) non-critical temperatures and (b) on an isotherm (i.e., −121.8◦C) near the critical isotherm. The data are offset vertically by the argon number density *ρ*Ar. The transition energy is *E*<sup>0</sup> = 8.424 eV for the unperturbed Xe 6*s* Rydberg transition.

## **4. Atomic perturbers**

#### **4.1. Xe low-***n* **Rydberg states in Ar**

#### *4.1.1. Xe absorption*

16 Will-be-set-by-IN-TECH

on the perturber/perturber and perturber/dopant ground-state intermolecular potential via *g*PP(*r*) and *g*PD(*r*). Thus, these simulations require one to develop a single set of ground-state and excited-state intermolecular potential parameters for each system. A

> *σ r* <sup>12</sup> − *σ r* 6

was chosen for the atomic perturber/perturber ground-state intermolecular interactions, and non-polar dopant/perturber ground-state intermolecular interactions. The ground-state molecular perturber/perturber intermolecular potential was a two-Yukawa potential, or

The ground-state polar dopant/perturber intermolecular interactions were modeled using a

− *σ*� *r* 6 <sup>−</sup> <sup>1</sup> *<sup>r</sup>*<sup>6</sup> *<sup>α</sup>Pμ<sup>D</sup>*

*α<sup>P</sup> μ<sup>D</sup>* 2 4 *ε*� *σ*�<sup>6</sup>

*α<sup>P</sup> μ<sup>D</sup>* 2 4 *ε*� *σ*�<sup>6</sup>

(The modified Stockmeyer potential includes orientational effects via an angle average that presumes the free rotation of the polar dopant molecule.) An exponential-6 potential, given

> 6 *γ e*

was chosen for the excited-state dopant/ground-state perturber interactions. In eqs. (41) - (44), *ε* is the well depth, *σ* is the collision parameter, *α<sup>P</sup>* is the perturber polarizability, *μ<sup>D</sup>* is dopant dipole moment, *χ* ≡ *r*/*re* (where *re* is the equilibrium distance), and *γ* is the potential

The Lennard-Jones parameters for the atomic fluids and the modified Stockmeyer parameters for the CH3I/Ar and CH3I/Kr interactions were identical to the parameters used to model accurately the perturber-induced shift of the dopant ionization energy for methyl iodide in argon [12–14, 61], krypton [12, 15, 61] and xenon [16, 61]. The parameters *κ*0, *z*1, *z*2, *ε* and *σ* in eq. (42) were adjusted to give the best fit to the phase diagram of the perturber [61].The parameters *ε*, *σ*, *χ* and *γ* in eq. (44) were adjusted by hand to give the "best" fit to the experimental absorption spectra of the dopant low-*n* Rydberg states in each of the fluids

2 ,

<sup>−</sup>1/6

*<sup>γ</sup>*(1−*χ*) <sup>−</sup> *<sup>χ</sup>*−<sup>6</sup>

.

<sup>12</sup>

which can be rewritten in standard Lennard-Jones 6-12 potential form [12], with

 1 +

1 − (6/*γ*)

<sup>e</sup><sup>−</sup> *<sup>z</sup>*<sup>1</sup> (*<sup>r</sup>* <sup>−</sup> *<sup>σ</sup>*) <sup>−</sup> <sup>e</sup><sup>−</sup> *<sup>z</sup>*<sup>2</sup> (*<sup>r</sup>* <sup>−</sup> *<sup>σ</sup>*)

, (41)

. (42)

<sup>2</sup> , (43)

, (44)

*V*(*r*) = 4 *ε*

*r* 

� *<sup>σ</sup>*� *r*

*ε* = *ε* � 1 +

*σ* = *σ*�

*<sup>V</sup>*(*r*) = *<sup>ε</sup>*

*<sup>V</sup>*(*r*) = <sup>−</sup> *<sup>κ</sup>*<sup>0</sup> *<sup>ε</sup>*

*V*(*r*) = 4 *ε*

standard Lennard-Jones 6-12 potential, or

modified Stockmeyer potential

by

steepness.

investigated here.

The Xe 6*s* and 6*s*� Rydberg states (where *s* and *s*� denote the *J* = 3/2 and *J* = 1/2 angular momentum core state, respectively) were experimentally measured in dense argon. As is discussed in Section 2.4.1, when Xe interacts with Ar, ground and excited state dimers form. These dimers are evidenced by blue satellite bands that arise on the higher energy side of the primary Rydberg transition. The Xe 6*s* Rydberg transition has two such blue satellite bands corresponding to eqs. (24)-(25), whereas the Xe 6*s*� Rydberg transition has a single blue satellite band corresponding to eq. (24) (cf. Fig. 2) [22–24]. The absence of the ground state XeAr dimer to XeAr eximer transition [i.e., eq. (25)] for the Xe 6*s*� Rydberg state may be caused by an extremely short lifetime preventing our ability to detect the transition or by the XeAr eximer decomposing during the excitation.

The solid lines in Fig. 8 present selected experimental Xe 6*s* Rydberg transitions doped into supercrtical argon at non-critical temperatures and along an isotherm near the critical isotherm offset by the argon number density. (Similar data for the Xe 6*s*� Rydberg transitions are not shown for brevity.) It can be clearly seen that the Rydberg transitions broaden as a function of the argon number density. The maximum of the absorption band also shifts first slightly to the red and then strongly to the blue, similar to the original observations of Messing, *et al.* [25, 26]. Since the ground state interaction between Xe and Ar (or XeAr and Ar) is attractive, the ground states are stabilized by the argon solvent shell. The slight red shift observed at low argon number densities indicates that the xenon excited states (either Xe∗ or Xe∗Ar) are also stabilized by the argon solvent shell. As the density increases, however, argon begins to shield the optical electron from the xenon cationic core, thereby decreasing the binding energy of the optical electron . Thus, as the density of argon increases the energy of the excited state also increases, leading to a blue shift in the transition energy at higher densities. Although not shown, the overall blue shift of the 6*s* Rydberg transition band is much larger than that of the 6*s*� band at the triple point liquid density of argon. This difference in overall shift is caused by the difference in the core state of the cation, since the *J* = 1/2 core state has a permanent quadrupole moment. This permanent quadrupole moment increases the interaction of the cationic core and the optical electron, thereby implying that the optical electron is less perturbed by the argon solvent shell.

However, since the blue satellite bands also broaden and shift with increasing argon density, the primary Xe transition becomes indistinguishable at medium to large argon number densities. Thus, the argon-induced energy shift of the primary Xe transition cannot be investigated directly using these data. Therefore, to probe perturber critical effects on the dopant excited states, we must first accurately simulate the absorption spectra over the entire argon density range at non-critical temperatures and on an isotherm near the critical isotherm.

#### *4.1.2. Discussion*

In order to simulate accurately the absorption spectra at high density, any line shape simulation has to include the primary transition, denoted *a* in Fig. 2 and given by eq. (23), as well as the two XeAr dimer transitions that yield the blue satellite bands, denoted *b* and *c* in Fig. 2 and given by eqs. (24) and (25), respectively. For the simulation of Xe in Ar, we chose to use eq. (41) for the ground state Ar/Ar, Xe/Ar, and XeAr/Ar interactions and eq. (44) for the Xe∗/Ar and Xe∗Ar/Ar interactions. We also required that the simulation use a single set of intermolecular potential parameters for the entire argon density range at non-critical temperatures and along the critical isotherm. All intermolecular potential parameters except the Ar/Ar ground state parameters were adjusted by hand to give the best simulated line shape in comparison to the experimental data. The values of these parameters are given here in Appendix A [37, 40].

The relative intensities of the simulated bands were set by comparison to the absorption spectra of Xe doped into argon at argon number densities where all bands could be clearly identified. Experimentally, at low argon number densities, the ratio of heights between the *b* band and the primary transition is 0.2 for both the Xe 6*s* and 6*s*� Rydberg states in Ar. For the Xe 6*s* Rydberg state in Ar, the ratio of heights between the *c* band and the primary transition is 0.45. Although for concentrated Xe systems, the ratio of heights for the blue satellite bands to the primary transition would increase with decreasing temperature or increasing perturber number density, this is not the case for the very dilute Xe/Ar system investigated here (i.e., [Xe] *<* 10 ppm for all argon number densities). Therefore, we can assume that the intensity ratio of the blue satellite bands to the primary transitions stays constant at different temperatures and different argon densities.

The dotted lines in Fig. 8 are the simulated line shapes for the Xe 6*s* transition at non-critical temperatures (cf. Fig. 8a) and on an isotherm (−121.8◦C) near the critical isotherm (cf. Fig. 8b). A similar figure for the Xe 6*s*� transition is not shown for brevity. Clearly, the simulated spectra closely match the experimental spectra for all densities. Both the simulated and experimental line shapes show a slight red shift at low argon number densities, followed by a strong blue shift at high argon number densities. With these accurate line shape simulations, moment analyses can be performed on the primary transition in order to investigate perturber critical point effects, as well as to discuss trends in solvation of different dopant electronic transitions in the same simple atomic fluid.

18 Will-be-set-by-IN-TECH

in overall shift is caused by the difference in the core state of the cation, since the *J* = 1/2 core state has a permanent quadrupole moment. This permanent quadrupole moment increases the interaction of the cationic core and the optical electron, thereby implying that the optical

However, since the blue satellite bands also broaden and shift with increasing argon density, the primary Xe transition becomes indistinguishable at medium to large argon number densities. Thus, the argon-induced energy shift of the primary Xe transition cannot be investigated directly using these data. Therefore, to probe perturber critical effects on the dopant excited states, we must first accurately simulate the absorption spectra over the entire argon density range at non-critical temperatures and on an isotherm near the critical isotherm.

In order to simulate accurately the absorption spectra at high density, any line shape simulation has to include the primary transition, denoted *a* in Fig. 2 and given by eq. (23), as well as the two XeAr dimer transitions that yield the blue satellite bands, denoted *b* and *c* in Fig. 2 and given by eqs. (24) and (25), respectively. For the simulation of Xe in Ar, we chose to use eq. (41) for the ground state Ar/Ar, Xe/Ar, and XeAr/Ar interactions and eq. (44) for the Xe∗/Ar and Xe∗Ar/Ar interactions. We also required that the simulation use a single set of intermolecular potential parameters for the entire argon density range at non-critical temperatures and along the critical isotherm. All intermolecular potential parameters except the Ar/Ar ground state parameters were adjusted by hand to give the best simulated line shape in comparison to the experimental data. The values of these parameters are given here

The relative intensities of the simulated bands were set by comparison to the absorption spectra of Xe doped into argon at argon number densities where all bands could be clearly identified. Experimentally, at low argon number densities, the ratio of heights between the *b* band and the primary transition is 0.2 for both the Xe 6*s* and 6*s*� Rydberg states in Ar. For the Xe 6*s* Rydberg state in Ar, the ratio of heights between the *c* band and the primary transition is 0.45. Although for concentrated Xe systems, the ratio of heights for the blue satellite bands to the primary transition would increase with decreasing temperature or increasing perturber number density, this is not the case for the very dilute Xe/Ar system investigated here (i.e., [Xe] *<* 10 ppm for all argon number densities). Therefore, we can assume that the intensity ratio of the blue satellite bands to the primary transitions stays constant at different

The dotted lines in Fig. 8 are the simulated line shapes for the Xe 6*s* transition at non-critical temperatures (cf. Fig. 8a) and on an isotherm (−121.8◦C) near the critical isotherm (cf. Fig. 8b). A similar figure for the Xe 6*s*� transition is not shown for brevity. Clearly, the simulated spectra closely match the experimental spectra for all densities. Both the simulated and experimental line shapes show a slight red shift at low argon number densities, followed by a strong blue shift at high argon number densities. With these accurate line shape simulations, moment analyses can be performed on the primary transition in order to investigate perturber critical point effects, as well as to discuss trends in solvation of different dopant electronic transitions

electron is less perturbed by the argon solvent shell.

*4.1.2. Discussion*

in Appendix A [37, 40].

temperatures and different argon densities.

in the same simple atomic fluid.

**Figure 9.** (a) The argon induced shift Δ(*ρ*Ar), as approximated by eq. (21), of the primary transition for the Xe 6*s* and 6*s*� Rydberg states as a function of argon number density *ρ*Ar at (•) non-critical temperatures and (◦) along an isotherm near the critical isotherm. (b) An expanded view of Δ(*ρ*Ar)The solid lines are a visual aid. See text for discussion.

A line shape analysis was performed on the accurate simulations of the primary Xe 6*s* and 6*s*� Rydberg transitions in order to determine the average argon induced shift Δ(*ρ*Ar) of the primary transition, as approximated from the first moment [i.e., eq. (21)]. This moment analysis is shown in Fig 9 as a function of reduced argon number density *ρr*, where *ρ<sup>r</sup>* = *ρ*Ar/*ρ<sup>c</sup>* with *<sup>ρ</sup><sup>c</sup>* <sup>≡</sup> 8.076 <sup>×</sup> 1021 cm−<sup>3</sup> [56]. Fig 9b shows an enhanced view of the perturber critical region with a critical effect in Δ(*ρ*Ar) clearly apparent. The absence of the red shift observed by Messing, *et al*. [25, 26] (cf. Fig. 3) results from our performance of a moment analysis on a blue degraded band, instead of a direct non-linear least square analysis using a Gaussian fit function on the primary transition. In other words, while the peak of the primary transition red shifts slightly at low argon densities, the first moment of the band does not, due to the perturber induced broadening.

General trends emerged in the behavior of the simulated line shape as a function of the intermolecular potential parameters. For instance, we observed that the strength of the asymmetric blue broadening of a band increases with increasing Δ*re* ≡ *r* (*g*) *<sup>e</sup>* <sup>−</sup> *<sup>r</sup>* (*e*) *<sup>e</sup>* [where *r* (*i*) *<sup>e</sup>* is the equilibrium dopant/perturber distance for either the ground state dopant (*i* = *g*) or the excited state dopant (*i* = *e*)]. However, the overall perturber-induced energy shift of the band depended on the ground state intermolecular potential well depth *ε*(*g*) as well as Δ*re*. The slight red shift at low perturber number densities, however, was controlled by the excited state intermolecular potential well depth *ε*(*e*) . Comparison of the Xe 6*s* and 6*s*� transition in Ar

#### 20 Will-be-set-by-IN-TECH 470 Advanced Aspects of Spectroscopy

shows that the 6*s* Rydberg state broadens and shifts to higher energies more quickly than does the 6*s*� state. Since both transitions are excited from the same ground state (implying that the ground state intermolecular potential parameters remain unchanged), Δ*re* must decrease and *ε*(*e*) increase in order to simulate the Xe 6*s*� Rydberg state in argon correctly. These general trends proved helpful when determining the intermolecular potential parameters for new systems.

Messing, *et al.* [25, 26] concluded that the argon induced energy shift is density dependent and temperature independent. However, both our experimental absorption spectra and the line shape simulations show a distinct temperature dependence near the argon critical point. To test the sensitivity of the perturber critical point effect, we extracted the perturber dependent shift Δ(*ρAr*) of the simulated primary Xe 6*s* transition in supercritical argon near the critical density along three different isotherms (i.e., *Tr* = 1.01, 1.06 and 1.11, where *Tr* = *T*/*Tc* with *Tc* = −122.3◦C). These data are shown in Fig. 10a and clearly indicate that the critical effect is extremely sensitive to temperature and can be easily missed if the temperature of the system is not maintained close to the critical isotherm.

**Figure 10.** (a) The calculated argon induced shift Δ(*ρAr*) doped into supercritical argon plotted as a function of reduced argon number density at a reduced temperature *Tr* = 1.01 (◦), 1.06 (•) and 1.11 (). (b) The local densities (*ρ*local = *g*max *ρ*bulk) of the first argon solvent shell around a central Xe atom plotted as a function of reduced argon number density at a reduced temperature *Tr* � 1.01 (◦), 1.06 (•) and 1.11 (). The solid lines are provided as visual aid.

If we return to the line shape equation [i.e., eq. (29)], we observe that the two-body interaction term *A*1(*t*) and the three-body interaction term *A*2(*t*) depend on the difference between the excited state and ground state intermolecular potentials and on the perturber/dopant radial distribution function. Since the potential difference will not depend dramatically on temperature, the critical point effect must be dominated by changes in the perturber/dopant radial distribution function *g*PD(*r*). In Fig. 10b, we plot the local density of the first solvent shell as a function of the bulk reduced argon number density on the same three isotherms. The *Tr* = 1.01 isotherm shows a much larger density deviation near the critical density in comparison to the other two isotherms. Thus, the argon induced blue shift is caused by the first perturber shell shielding the cationic core from the optical election. This increase in shielding decreases the binding energy of the electron, thereby increasing the excitation energy.
