CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

Theoretical Studies of Copper Clusters

A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Chemistry

by

Courtney Marie Sams

August 2013

The thesis of Courtney Marie Sams is approved:

______Dr. Daniel B. Curtis, Ph.D. Date

______Dr. Simon J. Garrett, Ph.D. Date

______Dr. Jussi M. Eloranta, Ph.D., Chair Date

California State University, Northridge

ii

Dedication

This thesis is dedicated to the four people that are not here now but meant so much to me in my life:

Jean Marie Fractious for giving me tenacity, Selma Sams for teaching me that being yourself is primary, Gregory Aaron Fractious for teaching me to be a good friend to all and Carol (Pat) Graham for always laughing at my randomness even when no one else would.

iii

Acknowledgements

I’d like to thank everyone who has helped me along on this journey:

Thank you Mom for passing on the IQ so I can (mostly) understand my research.

Thank you Dad for introducing me to the nerdiness which is so fundamentally me.

Jared (MB) for his amazing unflagging support even when he did not understand my ‘Sciency/Rain-man’ talk.

My numerous aunts and uncles who are there for me and my many cousins who knew me when all I wanted to do was read!

Thank you to my friends who stand by me and encourage me even when I am ready to give up and pursue other paths of life (Mojghan, Anna, Cassandra and of course, always Martha).

Thanks to the Department of Chemistry and Biochemistry staff (Dr. Kelson, Irene McGee and Migdonia (Sonia) Martinez) and the many other professors that had to put up with my method of expression.

Lastly, but not least, I’d like to thank my research advisor, Dr. Jussi Eloranta, for putting up with my need to work to pay for school in order to continue the research that I love.

iv Table of Contents

Signature Page ii Dedication iii Acknowledgements iv List of Tables vi List of Figures vii Abstract viii

1: Introduction to Computational Chemistry

1.1 Schrödinger Equation (time independent) 1

1.2 Born-Oppenheimer Approximation 3

1.3 Hartree-Fock 4

1.4 Coupled Clusters Method 8

1.5 Equation-of-Motion Coupled Clusters Method 10

1.6 Gaussian Basis Sets 10

1.7 Basis Set Superposition Error 12

1.8 Density Functional Theory 13

1.9 Accuracy of DFT Functionals 19

1.10 Relativistic Effects 20

2: Electronic structure of homonuclear diatomic molecules

2.1 Molecular orbitals 21

2.2 Optical Spectroscopy 26

3: Density Functional Theory Calculations of small copper clusters

3.1 Copper Dimers and Helium Impurities (CC-EoM) 33

3.2 Copper Clusters (DFT) 45

References 51 Appendix A: Journal of Physical Chemistry A, Vol. 115, pg. 1-34 (2011) 58

v List of Tables

Table 1: Comparison of performance of DFT methods by mean 19 absolute deviations

Table 2: Comparison of the performance of DFT methods with 20 respect to G2

Table 3: Electronic origins for ground and excited states of Cu2 40

Table 4: Equilibrium bond lengths of Cu2 (Å) 40

-1 Table 5: Harmonic frequencies for Cu2 (cm ) 41

Table 6: Parametrized potentials fitted to the form �(�) = 41

�� − �� − �� , in Å-cm units

-1 Table 7: Vibrational frequencies of Cu2 (cm ) 47

Table 8: DFT Optimized Copper - Copper Bond Lengths (Å) 48

Table 9: Binding energy He – Cu2 (meV) 49

vi List of Figures

Figure 1: Guide used to determine the order of atomic orbitals in 22 the overall molecular orbital diagram

Figure 2: Molecular orbital diagram for the F2 molecule 24

Figure 3: Visualization of bonding and antibonding molecular 25 orbitals

Figure 4: Molecular Orbital diagram for the excited F2 molecule 25

Figure 5: Flow chart utilized in point group determination 29

Figure 6: Direct Product table for the C3v point group 30

Figure 7: Images of thermomechanical He fountain 34

Figure 8: Thermomechanical He fountain schematic 35

Figure 9: LIF imaging of ablated Cu2 36

Figure 10: Visualization of copper dimer – helium interaction 39

Figure 11: Detailed view of potential energy curves of ground 42 state (X) and the six excited states for Cu2

Figure 12: Potential energy curve comparing ‘T’ and ‘L’ 44 configurations for Cu2 – He

Figure 13: Optimized copper cluster geometry 48

Figure 14: Visualization of Copper – Helium interaction 49

vii

Abstract

Theoretical Studies of Copper Clusters

A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Chemistry

by

Courtney Marie Sams

Master of Science in Chemistry

The ground and several excited states of copper dimer (X, A, B and C) were calculated by the coupled clusters equation-of-motion method. A qualitative agreement with gas phase spectroscopic observables, such as the electronic origins, harmonic and anharmonic vibrational frequencies and radiative lifetimes, were obtained. For the B and C states the calculations verify the experimentally generated potential energy curves for these states and further identify the weakly allowed A state to originate from the spin-orbit mixture of the B and A states. It was found that the present level of accuracy could only be reached when scalar relativistic corrections (Douglas-Kroll Hamiltonian; DKH) were included along with the DKH compatible triple zeta level basis set. Finally, the interaction between the copper dimer and atomic helium was calculated to elucidate the possible solvation structure of the dimer in superfluid helium. It was found that while Cu atoms are heliophobic, Cu2 dimers are heliophilic and should remain inside helium droplets. Using a pseudopotential method in the generalized gradient approximation and plane wave expansion, the binding energy and equilibrium bond lengths were calculated for Cu clusters from 2 – 4 and 7 atoms.

viii

ix

Chapter 1: Introduction to Computational Chemistry

1.1 Time Independent Schrödinger Equation

Erwin Schrödinger, an Austrian physicist and theoretical biologist, in 1926 proposed the wave equation as the foundation of quantum mechanics (Schrödinger’s equation):

(1) �  = E  where � is the Hamiltonian operator that represents total energy and Ei’s are the numerical values representing energy. Wavefunctions, i’s, contain all the general experimentally determinable information of the system. Eq. (1) has many possible eigenvalue and eigenfunction pairs that satisfy it. Mathematically Eq. (1) is called an eigenvalue problem. According to Bohr’s interpretation, ||2 is the probability density that when integrated over all space is normalized to unity (∫ ||dτ = 1). The

Hamiltonian operator in Eq. (1) includes the potential energy of the electron-electron repulsion, the nuclear-electron attraction and the nuclear – nuclear repulsion and the kinetic energy terms:

ℏ ∆ ℏ (2) � = − ∑ − ∑ ∆ − ∑ ∑ + ∑ ∑ + ∑ ∑ where the first and second terms are the kinetic energy of the nuclei and the electrons, respectively, the third term is the electron-nuclear attraction, and the fourth and fifth terms are the repulsions that occur between any electrons or any nuclei. N is the number of nuclei in the system, n corresponds to the number of electrons, the reduced Planck’s

1 -34 -31 constant is ℏ, equal to 1.054 x 10 J∙s, me is the mass of the electron (9.109 x 10 kg),

∆ (≡ � ) is the Laplacian operator that acts on the electron i coordinates, ZI is the

-19 atomic number for the Ith nuclei, e is the elementary charge, 1.602 x 10 C, and εo is the vacuum permittivity, which is equal to 8.854 x 10-12 F∙m-1.

It is currently not known how to solve the Schrödinger equation analytically for any system with more than one electron. As a result, approximations must be made.

Two such approximations are the Variational Principle and the Born-Oppenheimer (BO) approximation, discussed in Section 1.2. Although very different in nature, both have been used to help solve the Schrödinger equation for molecular systems. The Variational

Principle approximates the ground state of a system.

∗ ∫ (3) Ε = ∗ ∫ Typical application of the Variational Principle would be the following:

1. Choose � 2. Calculate Eref using � and Eq. (3) 3. Choose � 4. Calculate E using � and Eq. (3) 5. If E < Eref, {� = � and Eref = E}, Repeat step 3 if |E - Eref| > ε

where ε is some predetermined energy threshold. Ending this process is difficult because the actual Eo is not known. According to the Variational Principle, the trial energy, Etrial is found to be greater than or equal to the minimum energy, Eo (Etrial ≥ Eo). The

Hamiltonian operator, �, in Eq. (1) is hermitian which means that all eigenvalues and therefore also energies are real. Furthermore all eigenfunctions can be made orthogonal

∗ (∫ ψ ψ dτ = 0, when � ≠ �), and normalized (orthonormal). The eigenfunctions of Eq.

2 (1) form a complete basis set. A complete basis set means that it is possible to represent any function through a linear combination of these basis functions. A more detailed concept of basis function, will be discussed in Section 1.5.

1.2 Born-Oppenheimer Approximation

Solving Eq. (1) for a non – hydrogen-like molecule can be computationally expensive and extremely difficult if certain approximations are not utilized. One of these approximations is the Born-Oppenheimer (BO) approximation. The BO approximation separates the molecular wave function into the nuclear and electronic parts by assuming that the nuclei are fixed in place and only the electronic degrees of freedom need to be considered.1 The premise of this approximation is that the nucleus is so much heavier

(1800 times greater) than an electron and it can therefore be assumed to be stationary while the electrons orbit around the nucleus. In the BO approximation, the total wavefunction is written as a product of the nuclear and electronic parts

(4) Ψ(R, r) = Ψ(R) ∙ Ψ(R, r) where Ψtotal is the total wavefunction, Ψel is the electronic wavefunction, Ψn is the nuclear wavefunction, R corresponds to the nuclei and r to the electrons. This separation allows one to solve for the two wavefunctions independently. In order to solve for the electronic wavefunction, it is necessary to include the nuclear contribution as potential energy from the nuclei. In practical applications, the electronic wave function is solved first. After this, it is possible to solve for the nuclear wavefunction in a separate calculation. When utilizing the BO approximation in electronic structure calculations, the first term in Eq.

(2) is neglected and the fifth term is independent of the electronic coordinates. Although

3 BO approximation is useful in many ways, it may not be valid if two solutions with suitable symmetry are energetically close to each other.2

1.3 Hartree-Fock

In 1927 D. R. Hartree developed a method to approximate the energy for bosons

(e.g. 4He nuclei) which must have symmetric wavefunctions.3 Later V. Fock expanded upon this work and discovered that the Self-Consistent Field (SCF) method could be extended to antisymmetric wavefunctions, which is required for fermions (e.g. electrons).4 After Fock’s contribution, the method became the Hartree-Fock Method

(HF). HF is an approximation used in computational chemistry to determine approximate ground state wavefunctions and energies despite the complex nature of electron-electron repulsion. The HF method can give information about orbitals, electron density, and energy. Furthermore, it is possible to determine the equilibrium bond lengths – within

1% of the actual values, ionization energies – using the Koopman’s theorem (which states that the first ionization energy is equal to the opposite of the orbital energy of the highest molecular orbital), and harmonic force constants.5

The presence of electron-electron repulsion poses two distinct problems. The first is that separation of variables in the Schrödinger equation is not possible and the second is that large molecules have increasingly larger contributions from the electron-electron repulsion. The HF approach attempts to average electron-electron interactions by assuming that each individual electron is in the average electrostatic field of the other electrons instead of actually interacting dynamically with individual electrons. The downside of HF is that it does not directly treat electron correlation (i.e. electron-electron

4 interaction). Despite this deficit HF can be used for many chemical systems. Since HF belongs to the class of ab initio methods - meaning that no experimental data is utilized – it can also be employed to provide reliable initial molecular orbitals for more advanced methods that include electron correlation such as Configuration Interaction (CI), Møller-

Plesset (MP) or the Coupled Clusters (CC) approaches.5

Four assumptions intrinsic in the basic Hartree-Fock method are:

1. The Born-Oppenheimer approximation is valid,

2. All relativistic effects are negligible (e.g., spin-orbit coupling)

3. The orbitals can be expressed as a linear combination of a certain

number of basis functions,

and

4. The many electron wavefunction is represented by a single Slater

determinant, which does not include electron correlation.

An example of a Slater determinant for 2 electrons (He atom) is found below:

� (� ) � (� ) (5) �(� , � ) = √ �(�) �(�)

5 Where the electron coordinates, � = (�, �, �) and �(�) represent spinorbitals. For

example, in �(�), 1� implies that electron 1 is in the 1s orbital and has � spin. For fermions, it is necessary to account for the Pauli Exclusion Principle which states that two electrons cannot occupy the same spinorbital simultaneously. Therefore, the overall symmetry of the wavefunction in Eq. (5) must be anti-symmetric. As a result, for singlet states the spatial part must be symmetric and the spin part must be anti-symmetric, whereas for triplet states the spatial part must be anti-symmetric and the spin part

5 symmetric. Slater-type orbitals or orbitals from semi-empirical methods (e.g., Huckel) can be used as an initial guess for forming the first Fock operator. The HF equations are as follows:

(6) ��(1) = ��(1) ...... ��(1) = ��(1)

where � is the Fock operator, � is the spinorbital energy (the Fock operator eigenvalue), n is the number of occupied orbitals and � is the spinorbital eigenfunction. The eigenfunctions, �′s, in Eq. (6) are called HF spinorbitals since they also include the average interaction with the other electrons in the system. Eq. (6), used to calculate the

HF energy, is analogous to Schrödinger’s equation for one electron but it also depends on

the other spinorbitals. The Fock operator for a closed shell system, �, consists of the

core Hamiltonian, ℎ, Coulomb operator, �u, and the exchange operator, �u:

(7) � = ℎ + ∑ �(1) − �(1)

The core Hamiltonian, ℎ, includes the electronic kinetic energy and the nuclear – electron attraction for one electron (i.e., one electron operator). The Coulomb operator, �, is

∗ (8) �(1)� = � ∫ � (2) �(2)�� �(1)

-28 where � denotes a spinorbital (� ≠ �), � = which is 2.307 x 10 C∙V∙m and

� accounts for the classical electron-electron repulsion. �, the exchange operator, is defined as

6 ∗ (9) �(1)�(1) = �{∫ �(2) �(2)�� �(1).

Both � and � are two-electron operators. The HF equations can be solved using a multistep iterative process that eventually leads to converged spin orbitals (SCF; Self

Consistent Field). Practical numerical calculations employ the matrix form of the HF equations. The iterative process is as follows:

1. A set of spin orbitals are selected based on hydrogen-like wavefunctions or some

semi-empirical method such as Hückel (initial guess). Using these spin orbitals, a

set of related Fock equations are constructed.

2. The Fock equations are used to solve for new spin orbitals.

3. This process continues until a predetermined convergence is achieved. If no

convergence (e.g. |Ei – Ei+1| < ε, where ε is the convergence threshold) is

achieved it is necessary to go back to Step 2 and repeat the process.

A typical energy convergence threshold is 1 x 10-9 Hartrees, equivalent to 2.72 x 10-8 eV.

Fock operators are real and therefore also hermitian. The HF procedure yields m occupied spinorbitals and their corresponding energies. The remaining m-n unoccupied orbitals are called virtual orbitals. The orbitals can be plotted such that they can be identified, for example: 1s, 2s, … for atoms and σ, σ*, … for molecules. The equation for total energy (E) open or closed shell, is defined as:

(10) � = 2 ∑ � + ∑(2� − �) + �[2 ∑ � + � ∑(2�� − ��) + 2 ∑(2� − �)] where the first two terms correspond the closed shell part, and the bracketed terms are a combination of the open shell part and the open-closed interaction. The k and l indices refer to the closed shell electrons, m and n indices refer to the open shell electrons, a and

7 b correspond to different states of the same configuration -- dependent on the SCF program used and f is the fractional electron occupation of the open shell system.6

There are several variants of HF method: Restricted HF (RHF), restricted open- shell HF (ROHF), and unrestricted HF (UHF). The above section describes UHF except the energy calculation was for closed shell only (RHF). The RHF method is used for closed shell systems. ROHF is for systems with unpaired electrons but the spatial parts of the corresponding α and β orbitals are restricted to be identical.1

1.4 Coupled Clusters Method

The Coupled Clusters (CC) ab initio Method can be used to calculate approximate ground state electron correlation energies. Electron correlation is divided into two categories: dynamic and static7. Dynamic electron correlation is captured by generating typically single and double excitations from a single Slater determinant. The Slater determinant gives information about the wavefunction. The basic CC method can only treat dynamic correlation due to the complexity of the static correlation. The second type, static correlation has more than one dominant Slater determinant in addition to other insignificant Slater determinants (created by, for example, single and double electron excitations) used to calculate the correlation. Static correlation can be accounted for by the Multi-Configuration-SCF8 and MR-CI9 computational methods. The relationship between a single determinant HF wavefunction, Φ, and the corrected wavefunction is given as2:

̂ (11) Ψ = �̂ Φ

8 where �̂ ̂ is called the exponential excitation operator, which is equal to 1 + � + � + !

… + � (Mclaurin series) and is responsible for generating excitations from Φ . � is !

called the Cluster operator, where � = � + � + … + � and � is defined as the sum of the n-electron excitation operators. For example, a one electron excitation operator, which creates a singly excited determinant, would be represented by �Φ and a doubly

excited determinant could be ��Φ or �Φ. In this case �is equal to ∑ ∑ � ��

and � is equal to ∑, ∑, � ���� where i and j are occupied orbitals, a and b are

unoccupied orbitals, � is a coefficient, also called the single excitation amplitude that

indicates the weight of the Slater determinant, � , which is the double excitation

1 amplitude, � is the creation operator and � is the annihilation operator. The creation and annihilation operators are second quantization operators that allow for the removal

10 and creation of electrons in occupied and unoccupied orbitals. In the case of � an electron is being removed from occupied orbital i and placed into unoccupied orbital a.

In practical applications, the series is always truncated. For example, in the case of two

electrons, �̂ ̂ = 1 + � + �. !

The CC Singles and Doubles computational method (CCSD) is often used due to its ability to give almost exact result for electron correlation.11-13 The S in CCSD denotes one electron excitations and the D implies that two electrons are excited at the same time.

CCSD(T) implies that two electrons are excited but that the triple excitations are included approximately.1 The CCSD(T) method has been shown to give wavenumber accuracy for van der Waals dimers near the potential minima.14-17

9 1.5 Equation-of-Motion Coupled Clusters Method

Calculating excited state information is accomplished via the Equation-of-Motion

Coupled Clusters Method (CC – EoM).18 CC – EoM is based on a configuration interaction (CI) method used in conjunction with the Coupled Clusters method.

The CI method is based on the variational principle (Eqn. 3). The wavefunction used in

CC-EoM is as follows19

(12) Ψ〉 = R|Ψ〉 = R� Φ where Ψ is the excited state wavefunction, R is the CI type excitation operator and

19-20 Ψ is the coupled clusters wavefunction.

1.6 Gaussian Basis Sets

Numerical ab initio calculations require the use of a basis set to represent the wavefunction. A basis set is a set of functions that can be combined to represent a molecular orbital, �:

(13) � = ∑ �∅ where � are the expansion coefficients and ∅ are the basis functions. The basis set using

Slater-type orbitals (STO) is constructed as follows:21

(14) ∅(�, �, �) = �� � �(�, �) where � is the orbital exponent, �(�, �) is the spherical harmonic function, which determines the shape, N is the normalization constant, (�, �, �) represents a point in the spherical coordinate system centered on a given atom, ∅ is the basis function and n, l

10 and m are the principal, angular azimuthal and magnetic quantum numbers, respectively.

STO basis sets are not commonly used in computational chemistry as they are very expensive to evaluate numerically. Gaussian – type orbitals (GTOs) are more efficient basis functions to use in numerical calculations. In fact, you can use many GTO’s to approximate one STO and the calculation would still be significantly faster. This is because analytical formulas exist for integrating Gaussian functions. Cartesian GTOs are defined as:

(15) � = �� � � � where N is the normalization constant, x, y, and z are the Cartesian coordinates, t, u, and v are the exponents that are related to orbital angular momentum l = t + u + v , � is the

orbital exponent or best fit parameter, � represents the Gaussian part,8 and r is the distance from the origin. Note that there is a particular issues arising when using the

Cartesian representation. The Cartesian representation creates an extra orbital for l = 2 and above. This extra orbital does not appear when the spherical representation is used.12

For example, the d orbital in the Cartesian representation has one extra orbital that is linearly dependent on the others.

In order to approximate atomic orbitals using GTOs, it is often necessary to use contractions. Contractions are linear combinations of basis functions with fixed coefficients. Such contractions could include, for example GTOs combined in such a way that Slater type orbitals are obtained approximately. Thus the complete definition of a basis set consists of the exponents, azimuthal quantum numbers and contraction coefficients. Usually a shortened notation is used to describe basis sets such as STO-nG, where n = 3, 4, 5 or 6. STO-6G means that one STO is represented by 6 GTO’s. A

11 second commonly used type of basis set is called Correlation Consistent basis sets such as CC-VNZ, where CC denotes a correlation consistent, V implies a valence only basis set and N = D, T, Q, etc represents the number of basis functions used to replace the minimum basis set. The minimum basis set is the simplest type of basis set used to represent an orbital. There is one basis function (GTO) for each orbital in a given atom.

Double-zeta represents two GTOs per atomic orbital and triple-zeta is three GTOs per atomic orbital, etc. For example, there is one basis function used to represent both hydrogen (H) and helium (He). When using the CC-VTZ set, helium and hydrogen would be represented by three basis functions. In order to increase basis set effectiveness, or how well the basis set represents the species, it is possible to create an augmented set. These augmented basis sets have added diffuse functions or Gaussians that extend far away from the origin. Basis sets allow not only for the minimization of atomic energies, but the production of stable molecular structures.

1.7 Basis Set Superposition Error

The Basis Set Superposition Error (BSSE) is an error that arises from the application of an incomplete basis set.22 This error can be significant when calculating weak interactions such as van der Waals forces or hydrogen bonds in molecules. The most common methods to reduce BSSE are the chemical Hamiltonian approach (CHA) and the counterpoise method (CP). The goal of CHA is to account for BSSE during the actual calculation whereas CP attempts to remove BSSE after the calculation. The most commonly applied CP method is based on the procedure developed by Boys and

12 Bernardi.23 The magnitude of BSSE present can be estimated by studying the energy convergence as a function of the basis set size.24 Unfortunately, the computational cost required to do so is often highly prohibitive. In this work BSSE was minimized from the potential energy curves by the Boys-Bernardi method.23

If a complete basis set is used, the binding energy (ΔE) of a system is given by:

(16) Δ� = � − � − � where � is the energy of the dimer, � and � are the energies of separated atoms a and b, respectively. The Boys and Bernardi method utilizes ghost atoms, which have basis functions centered on them but no nuclear charge.25 The CP corrected energy is then given by

(17) Δ� = � − � − �

where ΔE is the corrected binding energy, � is the energy of the dimer, � is the

26 energy when b is a ghost atom and lastly, � is the energy when a is a ghost atom.

1.8 Density Functional Theory

1.8.1 Background

Density Functional Theory (DFT) expresses the total energy of a many particle system in terms of the one particle density.27 In electronic structure theory it can be used to solve, for example, ground state electronic energies and optimized molecular geometries.14 Until Kohn and Sham introduced the concept of orbitals to the DFT method, DFT was very inaccurate, in terms of the kinetic energy for non-interacting electrons.28-29 The use of Kohn-Sham (KS) orbitals allowed accurate calculation of the

13 kinetic energy for non-interacting electrons. Note that KS orbitals are only needed for evaluating this kinetic energy contribution because the coulomb, exchange and correlation contributions, which will be discussed below, can all be expressed in terms of the one electron density.

The general DFT starts with the assumption that the ground state electronic energy can be expressed in terms of the ground state one electron density only. From the computational chemistry perspective, pure DFT only requires storing one electron density

(� = �(�, �, �)) rather than the high dimensional many body wavefunction (� =

�(�, �, �, … , �, �, �)). Pure ab initio methods that focus on the wavefunction, instead of the one electron density, typically require a large number of variables to represent the many body wavefunction. In fact from the mathematical point of view,

DFT is very similar to HF. In contrast to the Hartree-Fock theory, DFT can be used to account for the missing electron correlation energy. Although small, the electron correlation energy is responsible, for example, for long-range van der Waals interactions.

Two drawbacks of DFT are that the accuracy of the calculation cannot be systematically improved and that the exact form of the energy functional is not known.

DFT expresses the total electronic energy through a model-dependent energy functional. Functionals are analogous to functions, in that a function maps a real or complex number to another number and a functional maps a function to a number. The

[] [] functional derivative of G[f], , is defined as lim where �� ||→ represents a change in the function.

Before the work of Walter Kohn and Le Jeu Sham DFT was not a practical computational method. Kohn and Sham hypothesized a hypothetical set of non-

14 interacting particles that gave the same density as particles that interact.27 For these particles, Kohn and Sham were able to develop a practical and effective potential, called the Kohn-Sham potential, which can be used to determine the exact ground state energy of a system with n electrons:

ℏ ∗ (18) �[�, �, … , �] = − ∑ ∫ � (�)∇ψ(r)dr − j ∑ ∫ ρ(r)d r +

12j0ρr1ρ(r2)r12 d3r1d3r2+ EXCρ

where ψ are the KS orbitals, � = Σ|�| , r1 and r2 represent electron coordinates, ZI is

-28 the atomic number of the nucleus, j0 = or 2.31 x 10 Jm, n is the number of electrons, N is the number of nuclei and �[�] is the exchange correlation energy. The second term in Eqn. (18) is the classical electron-nuclear Coulomb interaction and the third term is the corresponding interaction between electrons. Note that the electron- electron Coulomb interaction term is multiplied by ½ to avoid double counting. The exchange-correlation energy is defined as:

(19) � = (�[�] − �[�]) + (�[�] − �[�]) where (�[�] − �[�]) is the kinetic correlation energy and the (�[�] − �[�]) includes both the potential correlation and exchange energies. T[ρ] is the kinetic energy, T[ρ] is the kinetic energy functional, E[ρ] is the electron – electron repulsion and J[ρ] is the

Coulomb part.2

1.8.2 Local Density Approximation

Local Density Approximation (LDA) is one of the simplest exchange functionals that can be used in chemical applications and for example, for obtaining approximate molecular geometries. � is factored into exchange (Ex) and correlation (Ec) contributions:

15 (20) � = � + �

� is defined by LDA whereas the correlation functional, Ec, from the Vosko-Wilk-

Nusair or Perdew-Wang is typically used.

2 In LDA, the exchange energy functional � is written as :

(21) � [�] = −� ∫ �(�)��

where ρ is the electron density, C = , and the integration is carried over the three x

spatial degrees of freedom. The corresponding functional derivative, � , is given by:

(22) = − � � ≡ �[�]

Two major limitations to using LDA are:

1. Exchange energy is typically off by at least 10%, which is larger than a typical

correlation energy.

2. Assumes a slowly varying electron density, which may not be the case near

nuclei, for example.

Note that with wavefunction based methods, it is possible to improve the calculation accuracy by systematically adding more degrees of freedom to the wavefunction (e.g., including more Slater Determinants in the computation). Improving DFT requires development of a better functional.

The standard LDA method can only be used for closed shell systems, where

� = �. � is the α spin electron density and � is the β spin electron density. When considering open shell systems, it becomes necessary to account for the electron spin because � ≠ �. The DFT variant that accounts for this situation is called Local Spin

Density Approximation (LSDA). The difference between LSDA and LDA is that in

16 LSDA the α and β electron densities are stored separately. The LSDA exchange functional is defined as2

(23) � � , � = � [2� ] + � [2� ] = 2� [� ] + � [� ] where � = � + � is the total electron density. The corresponding LSDA functional derivatives are,

[, ] [] (24) = 2 [,] [] (25) = 2 . with respect to the ρα, and the functional derivative with respect to ρβ are given in

Equations (23) and (24).

The correlation energy, Ec, can be included in the calculation by Vosko, Wilk and

36 Nusair (VWN) or Perdew and Wang (PW) methods. The functional derivative, � , corresponding to VWN is:

() (26) �� , � = �(� , �) = � (� , 0) + � (� ) (1 − �) + [� (� , 0)]�(�)� "()

( ) ( ) ( ) where � = , � = �� , � = � + �, � � = and � and � ( )

2 are predetermined functions. The spin density, ς, is defined as � = � − �. Note that it is necessary to minimize energy functional with respect to both densities �and �, simultaneously.

The corresponding functional derivative for PW, �/ , is:

(27) �/ = −2��(1 + �� )ln 1 + (

2 where α, ρ, and β are model dependent parameters. ε and ε are determined by

() () (28) �⁄ (�) = � �� + ��� − �� + ��� () () ()

17

2 where x = �, X(x) = x + bx + c and Q = √4� − � . A, xo, b, and c are constants based on the LDA method utilized.

1.8.3 Generalized Gradient Approximation

Generalized Gradient Approximation (GGA) is an improved exchange correlation functional over LDA. Due to an increased accuracy, GGA is a bit more complex than

LDA and requires the electron density derivatives to be included in the functional. Two historically relevant exchange functionals used in conjunction with GGA are Perdew-

Wang (PW) 1986 and 1991 or PW86 and PW91.30

(29) � (�) = � (1 + �� + �� + �� )

|∇| where � = .

( ) (30) � = � ()

The most successful correlation functional was developed by Lee, Yang, and Parr (LYP).

(31) � = −4� − ���{ [144 2 � � + � + (47 − 7�)|∇�| ( ) 2 (45 − �)(|∇� | + ∇� | + 2�(11 − �)� ∇� | + � ∇� | + �(|∇� | − 3 |∇�| − |∇�| ) − �∇�| + �∇�| }

where � = and � = �� + , CF = (3� ) and a,b, c and ( ) ( ) d are parameters determined by helium atom data. Summing εc and εx gives the required functional derivative, εxc. Utilizing GGA allows one to obtain more accurate molecular geometry and energies than LDA. Despite accounting for the nonlocal interactions, GGA still does not allow for example, a description of van der Waals binding.31

18 1.9 Accuracy of the DFT functionals

Determining which computational method to use depends on the accuracy desired for a particular system. Generally most high level ab initio methods (e.g. G2) are more accurate than DFT.32 Within DFT, GGA based methods (e.g. B88, BPW91 and

B3PW91) are more accurate than LDA.31-32 As demonstrated in Table 1, the G2 ab initio method is most accurate for determining atomization energies, ionization potentials and proton affinities.32 Considering DFT, the LSDA (e. g., VWN) method is by far the least accurate as indicated by the mean absolute deviations in Table 1. These deviations are based on the experimentally determined atomization energies, ionization potentials and proton affinities. The lower the deviation, the more accurate the method used. The B88,

BPW91 and B3PW91 are all GGA based methods that although not as accurate as G2, show a deviation of almost an order of magnitude smaller than LDA.

Table 1: Comparison of performance of DFT methods by mean absolute deviations (x10 kJ/mol)33 Method Deviations in Deviations in Deviations in Atomization Energy Ionization Potential Proton Affinity G2 0.5 0.6 0.4 LSDA 14.9 2.6 2.3 B88 1.6 4.7 1.0 BPW91 2.4 1.7 0.6 B3PW91 1.0 1.6 0.5

The samples utilized to determine the data in Tables 1 and 2 include hydrocarbons, substituted hydrocarbons (chlorine and fluorine), inorganic hydrides and carbon and oxygen centered radicals. As seen in Table 2, G2 had the least deviation from experimental values for the cyclic hydrocarbons with unsaturated rings (0.8 – 1.6 kJ/mol). The B3LYP, a GGA method, was the best DFT method while BLYP was the

19 most accurate of the LSDA based method. These assumptions are based on the preferred chemical accuracy of 4 kJ/mol.34

Table 2: Comparison of the performance of DFT methods with respect to G2 (x10 kJ/mol)33 Method Mean absolute deviation Maximum absolute deviation G2 0.7 3.4 SVWN 38.0 95.6 BLYP 3.0 11.9 BPW91 3.3 13.5 B3LYP 1.3 8.4 B3PW91 1.5 9.1

1.10 Relativistic Effects

For many chemical systems, which include elements from the first three rows of the periodic table, it is often possible to ignore relativistic effects.2 The most important relativistic effects, relevant to chemistry, include electron spin, spin-orbit coupling (SOC) and scalar effects. Scalar effects are responsible for the contraction of the inner core orbitals. The latter can be included in the calculation by the Douglas-Kroll – Hess

(DKH) Theory.35-40 SOC is a weak interaction between the spin of the electron and the orbital angular momentum.41-42 Its magnitude is characterized by the spin-orbit coupling.

This can cause splitting of certain degenerate states. For simplicity, consider the 2p orbital frame in the boron atom, where the spin orbit coupling is 16 cm-1.43

20

Chapter 2: Electronic structure of homonuclear diatomic molecules

2.1 Molecular Orbital Theory

Approximate molecular orbitals (MOs) for diatomic molecules can be constructed from a linear combination of the atomic orbitals (LCAO). Consider, for example, the H2 molecule with atomic orbitals 1s,A and 1s,B centered on nuclei A and B, respectively.

The two possible (MOs) in this case are:

(32) � = (� + � ) √ , ,

(33) � = (� − � ) √ , ,

1 where � is the bonding MO and � is the antibonding MO. When the highest occupied molecular orbital (HOMO) is a bonding orbital, the bond between atoms is strong. When HOMO corresponds to an anti-bonding orbital, the bond is weak and the

21

Figure (1): Guide used to determine the order of atomic orbitals in the overall molecular orbital diagram.

molecule is unstable or very reactive. This can be expressed through bond order, which is defined as

(# �� ��������� �� ������� �������� − # �� ��������� �� ����������� ��������).

22 Note that for heteronuclear diatomic molecules, only the atomic orbitals with the same symmetry and similar orbital energies mix to form valid molecular orbitals.44 For diatomic molecules with more than two electrons, many MOs will be occupied. The MO building order based on orbital energies is given in Figure (1). Electrons are added to the

MOs using the Aufbau Principle, Pauli Exclusion Principle and Hund’s rules (for molecules). The Aufbau Principle states that electrons fill from lower energy levels to higher energy levels. The Pauli Exlusion principle states that no two electrons can have the same set of four quantum numbers. Hund’s rule states that the ground configuration with the maximum multiplicity (2S+1) usually lies lowest in energy. Based on given bond length of 2.7 Å for Li2, the Cu2 molecule is most likely located in the same position as the Li2 molecule in Figure (1) as their bond lengths are the most similar of the molecules mentioned. The H2, Li2, C2, O2, N2, F2, B2 and Ne2 bond lengths are 0.7, 2.7,

1.2, 1.2, 1.1, 1.4, 1.6 and 5.8 Å respectively.45-48

Consider, for example, the MO diagram for the F2 molecule (point group D∞h) as shown in Figure (2). Each MO made of s-orbitals is either σg or σu, where σg corresponds to a bonding and σu to the antibonding orbital (cf Eqns (32) and (33)). A graphical presentation of the MOs for F2 is given in Figure (3). When an electron is excited from one MO to another, an excited state molecule is formed. For example, if one of the electrons in F2 is excited from πg to σu, Hund’s rule is violated and the molecule is in the excited state. The new MO diagram for this excited molecule is shown in Figure (4).

23

σu

πg 2pA 2pB πu

σg

σu

Energy 2sA 2sB

σg

σu

1sA 1sB

σg

Figure (2): MO diagram for the electronic ground state of F2 molecule.

24 a

b d

c

Figure (3): Formation of bonding and antibonding molecular orbitals from a. two 1s atomic orbitals. b. 2px,A and 1sB (no bonding). c. 2pz,A and 2pz,B. d. 2px,A and 2px,B.

Energy

Figure (4): MO diagram for the electronically excited F2 molecule where the electron is promoted from the πg orbital to the σu orbital.

25 2.2 Optical Spectroscopy

Electronic structure of atoms and molecules can be studied by optical spectroscopy based experimental methods. Such measurements yield information about the electronic transitions induced by photons in terms of spectroscopic line positions

(transition energy) and intensities (transition moment). The transition moment is related to the absorption line intensity through the Fermi golden rule49 or to the radiative lifetime in light emission measurements (fluorescence or phosphorescence). The Fermi Golden

Rule is an equation used to calculate transition rates. The important observables in such experiments are the transition energy and the corresponding line intensity of two electronic states.

2.2.1 Molecular Term Symbols

The electronic states of diatomic molecules are labeled by molecular term symbols, which include information about spin multiplicity, total orbital angular momentum about the internuclear axis and the total amount of angular momentum. The

S general form of a molecular term symbol is Λ. The total orbital angular momentum about the internuclear axis quantum number, Λ, is obtained as a sum over the orbital angular momenta of the individual electrons:

(34) Λ = |� + � + ⋯ + �| where � is the angular momentum quantum number for electron i. � = 0 represents a σ orbital, � = ±1 a π orbital, ±2 a δ orbital, etc. The values of Λ are labeled according to Λ = 0 (Σ), Λ = 1 (Π), Λ = 2 (Δ) and Λ = 3 (Φ), etc. The spin multiplicity is expressed by 2S + 1, where S represents the total electron spin (e.g., 3 for triplet, 2 for doublet and 1

26 1 for singlet). For example, the molecular term symbol for ground state F2 is Π (Figure 2) and 3Σ for the first excited state shown in Figure 4. The above molecular term symbol classifications are valid when spin – orbit interaction is small (Hund’s case (a)).

However, when spin – orbit interaction is large, S and Λ are no longer good quantum numbers (Hund’s case (c)). Instead the sum J = |S + Λ| must be used to characterize the states.29 Computations utilize MOs in order to allow for an overall faster set of calculations.

If a homonuclear diatomic molecule has a center of symmetry, MO functions may change sign (e.g., even vs. odd) when inverted through the center of symmetry. For even parity (�. �. �(�, �, �) = �(−�, −�, −�)) the MO is labeled g whereas for odd parity,

(�. �. �(�, �, �) = −�(−�, −�, −�)) u is used. The overall u/g label for the term symbol is obtained by considering the even/odd symmetries for all occupied MOs and using the following multiplication rules:

g · g = g, g · u = u, u · g = u and u · u = g

Orbitals can be further classified using + and – by considering reflection through the plane that is parallel to the bond axis. The +/– classification is only valid for Σ states.

When an orbital has even symmetry with respect to this operation a “+” is used and a “-” is used when it is odd. An example of a complete term symbol for ground state O2, using

1 the g/u and +/- classification systems is Σ .

2.2.2 Group Theory

Selection rules are a significant component of the spectroscopy of a molecule.

These selection rules are derived from integrals obtained from allowed and forbidden transitions (i.e., transition dipole moment). In this section we focus on using group

27 theory to evaluate the integrals that appear in molecular spectroscopy. Ab initio calculations can be used to reduce the number of basis functions required in the calculation since they can be obtained by the relevant matrix forms of the symmetry operations.

A symmetry operation is defined as any operation that rearranges the nuclei of a molecule in such a way that the molecule does not change. The five symmetry operations are identity (E), n-fold rotation (Cn), relection (σ), inversion (i) and n-fold improper rotation (Sn). The E operation leaves the object or molecule unchanged, Cn is a counter- clockwise rotation around an axis by 2π/n, σ is a reflection operation in the vertical or horizontal plane. σv includes the principal symmetry axis and σh is when the principal symmetry axis is perpendicular to the mirror plane, i is an inversion operation through the center of symmetry and Sn is a n-fold rotation (Cn) followed by a horizontal reflection through the mirror plane.1

A point group is defined as all the symmetry elements a molecule may contain.24

Using the symmetry operations for a given point group, it is possible to deduce the point group of a molecule. Figure (5) gives a sample flow chart to be used in the determination of a molecules point group. For example, benzene has a point group of D6h.

28

Figure (5): Flow chart utilized in point group determination. The definition of a group is:

1. The identity (E) operation belongs to the group.

2. The elements multiply associatively. For example, (RS)T = R(ST).

3. If R and S are elements, then RS also belongs to the group.

4. The inverse of each element is a member of the group (i.e., RR-1 = R-1R= E).

Symmetry operations can be multiplied, for example, RS corresponds to an operation with S first and then with R. Additionally, RS is only equal to SR if they commute (RS –

SR = 0 also called an Abelian group). Symmetry operations can also be expressed in

matrix form. An example of a matrix representation of the � symmetry in C3v point group for NH3 is:

1 0 0 0 0 0 0 1 D(�) = 0 1 0 0 0 0 1 0

29 The basis set in this example consists of four 1s orbitals centered on each atom as shown below:

H sc

N H sn H sa sb

The orbitals are ordered as follows: sn (N), sa (H), sb (H), sc (H). Once a matrix representation is found, it is possible to reduce the four dimensional matrix representation into a three dimensional representation by considering only (sa, sb, sc) as follows:

0 0 1 D’(� ) = 1 0 0 0 1 0

The above change in dimensionality is called a reduction which is only possible if a diagonal form for the given basis is found. When no more reductions are possible, the representation remaining is considered an irreducible representation (irrep).

When a complete list of characters, the diagonal sum of the matrix elements, is found for the molecules, a character table can be constructed.

A A E C3v 1 2

A1 A1 A2 E

A2 A2 A1 E

E E E A1 + A2 + E

Figure (6) Direct Product table for the C3v point group.

30 Next we demonstrate how to apply group theory to evaluate integrals that frequently appear in quantum mechanics. For example, the irreps for the functions appearing in overlap integral, ∫ ����, can be assigned within a given point group. If for example, f1 belongs to A1 and f2 to B1, within C2v, the direct product gives � · B1 = B1.

In general, if the direct product is not equal to the completely symmetric irrep, � �� �

(depending on the point group), the integral is equal to zero. The same method can be applied to ∫ �����, where the direct product would give � � � � �, if f3 belongs to

A1. Determining the direct product would lead to �� � which would equal �. Once again, the integral is equal to zero. This integral with three functions is relevant for the derivation of selection rules where f1 would correspond to the initial electronic state, f2 to the electric dipole moment operator, and f3 to the electronic final state.

2.2.3 Molecular Selection rules

Determination of allowed (i.e., non-zero line intensity) vs. forbidden (i.e., zero line intensity) transitions is based on the Fermi golden rule49 and the evaluation of the transition dipole moment between the electronic states under consideration. The transition dipole moment is defined as

∗ ⃗ (35) �⃗ = ∫ ��̂��� where � is the electric dipole moment operator, which is proportional to the electron coordinates x, y, z, �⃗ is the transition dipole moment (Cm or Debye) and � and � are the wave functions for the initial and final electronic states.50

31 Selection rules can be derived using group theory as described earlier. For example, for homonuclear diatomic molecules in D∞h, one can assign irreps to

⃗ �, � and �̂. It is then possible to use the appropriate direct product table to see if Σ is obtained. The applicable selection rules for diatomic molecules are as follows:

1. ΔΛ = 0, ±1, for example Σ – Σ, Π – Σ is allowed but Σ – Δ is forbidden. 2. ΔJ = 0, ±1, where J is the total angular momentum and equal to |S + Λ| 3. Σ+ – Σ+ and Σ- – Σ- are allowed transitions although, Σ+ – Σ- is not. 4. g – u is allowed but g – g and u – u are forbidden 5. ΔS = 0 therefore a singlet – singlet transition is allowed but singlet to triplet is forbidden. This may breakdown when spin-orbit interaction is large. 6. ΔΣ= 0, where Σ = –S, –S+1, …, S – 1, S

32

Chapter 3: Electronic Structure Theory Calculations of Copper

3.1 Cu2 – He atom

Background

The most important experimental methods to study the interaction of atoms/molecules (“impurities”) with superfluid helium are based on the helium droplet technique51 or the bulk helium approach.52-53 The helium droplet experiments, especially, have provided a wealth of spectroscopic information, which require high-level theoretical calculations to interpret the data. The most theoretical methods for describing superfluid helium are based on variations of the Monte Carlo method (e.g., variational and diffusion

Monte Carlo) or bosonic density functional theory (Orsay-Trento functional).54-55 Both of these methods rely on the accurate description of the pair potential between the impurity and ground state helium atoms. In this work, accurate ab initio calculations of

Cu – He, Cu2 and Cu2 – He are presented. The first and last systems are directly relevant for the bosonic DFT calculations carried out in Professor Eloranta’s group, whereas the copper dimer calculations were used to assess the accuracy of the applied ab initio method.

33

Figure (7): Images of the thermomechanical helium fountain. Images show varied operation modes (small blob, large blob, misty mode and nondivergent column flow). The fountain diameter is 200μm.

For the Eloranta laboratory, the experimental goal was to determine the interaction of Cu2 with bulk superfluid helium. This determination is accomplished via

Laser Induced Fluorescence (LIF). LIF is a spectroscopic method used to study the structure of molecules, where a laser is used to ablate the metal target (Cu). The copper atoms and copper dimers come off the target and are either reflected from or go into the thermomechanical helium fountain, as seen in Figure (7). LIF imaging gives insight into the presence of Cu2 in or near the thermomechanical helium fountain as seen in Figure

(8). Depending on the impact velocity of Cu/Cu2 to the superfluid helium fountain surface, they either get reflected off of or solvated into the fountain.

34

P ~ 10 torr (He)

Laser

Lens

Cu target

Heater (made of porous material)

Superfluid helium bath

Figure (8): Thermomechanical helium fountain schematic. Body of fountain surrounded by superfluid helium leading to the “fountain effect.”

The thermomechanical fountain utilizes the heater to pump the helium from the

superfluid helium bath out and through the top of the fountain. The laser shines through

a lens and focuses down onto the Cu target. The Cu atoms and Cu2 molecules

accumulate in between the target and the fountain. Depending on their

heliophilicity/heliophobicity as well as the velocity, the atoms/molecules will enter the

helium fountain or will stay in the area between the fountain and the target. It was

determined by the Eloranta laboratory group that there is a threshold velocity required for

Cu2 to enter the helium fountain as evidenced in Figure (9). Note that at a velocity of 1.5

35 m/s, the majority of the Cu2 gas is found in between the target and the fountain.

Alternatively, when the velocity is equal to 15 m/s, the majority of the Cu2 gas is found in the helium fountain, represented by the blue line in Figure 9.

Figure (9): LIF imaging of ablated copper dimer. Blue line represents the helium fountain, the red line represents the copper target and the white horizontal area indicates the presence of copper dimers.

To develop an accurate picture of Cu and Cu2 interacting with superfluid helium, ab initio calculations of Cu – He, Cu2 and Cu2 – He are described in the following sections. By combining both experimental and theoretical techniques, it is possible to determine the interaction of Cu and Cu2 with the superfluid helium fountain. Developing an accurate picture of the Cu – He and Cu2-He potentials requires a valid ab initio calculation of Cu, Cu2 and Cu2-He interaction as seen in the following sections. Utilizing the LIF methods, bosonic DFT and ab initio calculations it is possible to reach the computational goal of determining the locations of the copper atom and the copper dimer in regards to heliophobicity and heliophilicity, where heliophobicity is the tendency of

36 the copper dimer to stay out of the helium fountain and heliophilicity is the tendency to travel into the helium fountain

Method

We investigated the electronically excited states of the copper dimer (Cu2) and its interaction with one helium atom by the Coupled Cluster Equation of Motion (CC-EoM) method (Figure 11). The studied excited electronic states lie above the ground up to 6 eV

(λ > 200 nm). The interaction between two copper atoms was first modeled using the

Restricted Hartree-Fock (RHF) and then followed by Coupled Cluster Singlet Doublet –

Equation of Motion (CCSD - EoM) calculation. For the calculation to be accurate a relatively large basis set had to be used. The basis set utilized in this work was aug-cc- pwCVTZ-DKH54, where the ‘aug’ means that the basis set has been augmented (i.e.,

Gaussians with small exponents), ‘pw’ implies spin polarization added, ‘cc’ means correlation consistent set and DKH that the basis set is compatible with the relativistic

DKH formalism. Due to the well-known molecular dissociation problem of RHF55 that

+ - leads to the breakdown of a Cu2 molecule into Cu + Cu versus the Unrestricted HF Cu –

Cu required to calculate data, only nuclear distances near Cu2 equilibrium bond length were considered. The lowest lying molecular states of Cu2 were studied in experiments at both CSUN and UC Irvine, recently.59 The relevant ground and several excited states

1 3 1 1 of copper dimer (X ( Σg), A( Πu), B( Σu) and C( Πu)) were considered. A qualitative agreement with gas phase spectroscopic observables, such as the electronic origins, harmonic and anharmonic vibrational frequencies and radiative lifetimes, were obtained as shown in Tables (3 - 6). For the B and C states the calculations confirm the experimentally determined potential energy curves for these states and further identify the weakly allowed A – X transition to originate from the spin-orbit mixture of the B and

37 A states (singlet-triplet mixing). It was found that the present level of accuracy could only be reached when scalar relativistic corrections (DKH) were included along with the compatible triple zeta level basis set.

The interaction between the copper dimer and atomic helium was calculated to elucidate the possible solvation structure of the dimer in superfluid helium. Such calculations can be carried out using bosonic DFT.59 As a result, the first step of analysis was to construct a potential energy curve for a ground state copper dimer, Cu2, with one helium atom. The calculation of copper dimers interacting with bulk superfluid helium is not a trivial task. This requires modeling of superfluid helium (4He) which consists of strongly interacting bosons. We assumed two possible configurations for the copper dimer – helium interaction, as seen in Figure (9). Other possible configurations were discounted as insignificant. The Cu2 can have either a helium atom perpendicular to the bond or directly attached to one of the coppers in the dimer in a linear configuration.

When the helium atom is perpendicular to the Cu-Cu bond the configuration is denoted as the 'T' configuration. Conversely, the 'L' configuration (linear) occurs as a result of the helium atom placed in line with the Cu – Cu bond. The overall potential for Cu2 interacting with liquid helium can then be obtained by assuming pairwise additivity.

38 2.18

4.34

2.18 3.43

Figure (10): Visualization of copper dimer – helium interaction. The copper atom is represented by the blue ball and the helium atom is represented by the white ball. a. ‘T’ configuration of the copper dimer interation with one helium atom. The bond lengths between the coppers is 2.18 Å The bond length between the copper atoms and one helium atom is 4.34 Å. b. ‘L’ configuration of the copper dimer interaction with one helium atom. The bond lengths between the helium and the copper dimer is 3.43 Å. Notice that the Cu – Cu bond is relatively close to the experimentally determined bond length of 2.2 Å.

Results

The electronic origin (Te), equilibrium bond lengths (Re), harmonic frequency (ωe) and the radiative lifetime were calculated in this study by CC-EoM. The spectroscopic

-1 constants were calculated for Cu2 only. Electronic origin is the distance in cm , the equilibrium bond length is for the excited states A, B and C from the ground state X.

Equilibrium bond length, in Å, is the minimum energy distance between the atoms in the

Cu2 molecule at different energies for different states. The harmonic frequency in diatomic molecules indicates how steep or shallow the potential shape. The harmonic frequency was determined by evaluating the force constant based on the second

59 derivative of the potential energy curve. In each table below, Te(Exp) is the

11,62-66 experimental data determined from the Eloranta laboratory group. Te(Exp) is data from outside sources and lastly, Te(CC-EoM) is direct computational research produced for this thesis. The resulting data is in Tables (3 – 6):

39

-1 Table (3): Electronic origins for ground and excited states of Cu2 (cm ). 59 11,62-66 State Te (Exp.) Te (Exp.) Te (CC-EoM)

X 0 0 0

A -- 20431 20000

B 21767 21758 22260

C 21875 21866 22660

In Table 3, electronic origins of ground state X and excited states, A – C, are given in

-1 wavenumbers (cm ). Te is simply the graphical location of states A – C in comparison to the ground state X.

Table (4): Equilibrium bond lengths of Cu2 (Å)

59 11,62-66 State Re (Exp.) Re (Exp.) Re (CC-EoM)

X 2.22 2.22 2.25

A -- 2.28 --

B 2.32 2.23 2.38

C 2.29 2.26 2.25

In Table 4, the equilibrium bond lengths are given for ground state X and excited

59 states A – C in Å. In the above table, Re(Exp) is based on the Eloranta laboratory data.

11,62-66 Re(Exp) , is previous experimental data completed by Rohlfing, et. al,

Bhattacharyya, et. al, Page, et. al, Lochet, Ram, et. al and McCaffrey, et. al. Re(CC-

EoM) is my personal computational data used for this thesis.

40

-1 Table (5): Harmonic frequencies for Cu2 (cm ) 59 11,62-66 State ωe (Exp.) ωe (Exp.) ωe (CC-EoM)

X 275 266.5 251

A -- 192.5 --

B 244±1 246.3 216

C 218 221.4 240

Table 5 allows for the comparison between the harmonic frequencies of the

59 11,62-66 Eloranta laboratory data (ωe (Exp.) ), previous experimental data (ωe (Exp.) ) and my computational calculations (ωe (Theoretical)).

Table (6): Parametrized potentials fitted to the form: �(�) = �� − �� − -1 �� , in Å-cm units. Rmin (in Å) and Vmin (in cm ) correspond to the minimum energy distance and energy, respectively. T and L denote the linear and T-shaped (perpendicular) geometries59.

Cu – He L – Cu2(X) – He T-Cu2(X) – He L – Cu2(B) – He T – Cu2(B) – He

6 6 6 6 ao 1.1 x 10 5.0 x 10 2.1 x 10 3.6 x 10

a1 2.544 2.446 2.590 1.973 2.814

4 5 5 a2 7.661 x 10 7315 1.236 x 10 39.81 1.326 x 10

6 7 6 7 5 a3 1.079 x 10 1.383 x 10 1.723 x 10 3.357 x 10 8.878 x 10

rmin 4.8 5.0 4.8 7.2 4.1

Vmin 4.6 11.7 7.9 ~2 17.8

The minimum energy geometries for the T and L – shaped orientations are shown in Table 6.

41

Figure (11): Calculated potential energy curves of ground state (X) and the six excited states for Cu2. The arrows indicate B – X and C – X transitions and there corresponding radiative lifetimes. Note: scale is in eV to reflected in the very large energy differences between states.

Utilizing Ancilotto’s criterion67 for heliophilicity and heliophobicity – where heliophilicity is defined as solvation into helium and heliophobicity is defined as repulsion from helium – we are able to make a connection between the copper atom and dimer interaction with helium. Based on the Ancilotto criterion, Cu atoms are predicted to be heliophobic whereas Cu2 appears to be an intermediate case, which requires a more detailed bosonic DFT calculation to evaluate its solvation in superfluid helium.

Discussion

Due to the well-known ionic dissociation problem of RHF60, one might expect that

Unrestricted HF would provide better reference orbitals. However, both failed the

42 68-69 standard CC T1-normal test at large bond lengths. This test essentially indicates the degree of static correlation present in the wavefunction. Despite the wealth of experimental spectroscopic information for copper dimers70, theoretical calculations of the excited states have not been carried out successfully before.12 Thus the present results offer a significant improvement over the previous theoretical studies.71-72

The following RHF based closed shell equation-of-motion (EoM) coupled clusters (CC) method with single and double excitations (SD), converged quickly and without problems to the excited states near the equilibrium geometry. This is in contrast to the standard CI based methods that experienced convergence issues for various different reasons. Thus the development of CC-EoM method was essential for obtaining the present data for Cu2. In a separate calculation, the spin-orbit coupling was included for the excited states A and B, allowing it to mix them. Based on the selection rules, the

3 1 Πu (A) to Σ (X) transition is forbidden. However, the spin-orbit coupling allows for the mixing of the singlet and triplet states, such that the ΔS = 0 selection rule breaks down. This was confirmed experimentally in Reference 59. The degree of mixing can be obtained by diagonalizing the following spin-orbit Hamiltonian matrix:

� � �( �) − 2 √6 2 ⎛ ⎞ (36) ⎜ � ⎟ √6 2 �( � ) ⎝ ⎠

3 where �( �) and �( � ) are energies at a given bond length, Πu represents the A state

1 + and Σu represents the B state. The spin-orbit coupling of the copper atom is

43 represented by ξ (0.1 eV).59 Inspection of the eigenvectors revealed a significant degree of mixing indicating that the X – A transition became allowed.

Figure (12): Potential energy curve comparing ‘T’ and ‘L’ configurations for Cu2 – He. Note that Energy scale is in Kelvin to highlight the very small solvation energies.

To determine the ground state potentials for the Cu – He and Cu2 – He calculations, iterative triples were included to the normal CCSD method to increase accuracy of the electron correlation treatment (CCSD(T)). The CCSD(T) method and a large basis set, aug-cc-pV5Z (AV5Z), allowed for the accurate calculate ion of van der

Waals interaction where it is important to also account for BSSE.73 The actual calculation was carried out for He – Cu2 – He supermolecule (D2h) and the resulting energies were divided by two to give the Cu2 – He (C2v) interaction. This assumes that the two He atoms in the supermolecule are far away from each other and do not interact.

Radiative lifetimes were found to be 200 ns for the B state and 1 μs for the C state, compared to the experimental values, which were 20 ns and 1 μs.59 The small difference

44 between the experimental and theoretically calculated lifetimes is most likely due to the limited basis set used in the present work.

The potential energy calculated in Figure 12 allows one to compare the minimum bond lengths for both the Cu2-He T and L calculations. This data looks at both the ground state, X, and one excited state, B, in addition to the potential energy (in Kelvin) for the Cu-He interaction. Note that the Cu2-He L approach is more attractive than any other ground state interaction, including the Cu-He interaction, as evidenced by the depth of the potential energy curves. For example, the Cu – He interaction has the least attractive/most repulsive interaction, which confirms the heliophobicity displayed when copper comes into contact with the helium fountain.

3.2 Cun - He (DFT)

Background

Metal clusters are important in surface chemistry and heterogeneous catalysis.74-79

A limited amount of research has been focused on the copper cluster and even less for the interaction between the copper dimer and helium bath. Knickelbein and Powers, et. al. focused on the ionization of ‘n’ number of copper atoms (n = 2 – 150 and n = 1 – 13, respectively)56-57 while Pettiette, et. al.58 researched the interactions of copper in a gaseous helium (n = 6 – 41). In this study, small copper clusters with n = 2 – 4 and 7 atoms were studied by DFT as a method of evaluating the DFT and van der Waals interaction. Their energies and equilibrium geometries were obtained and the validity of

DFT in describing van der Waals interaction in He – Cun was assessed .

45 Method

This study is based on quantum mechanical calculations within the framework of the electronic DFT. All the calculations are based on pseudopotential basis set description within DFT in the generalized gradient approximation (GGA using the Perdew-Burke-

Ernzerhof (PBE) exchange correlation functional.2,81) The basis set was based on plane wave expansion of the electron wave functions.82 Pseudopotentials act by turning the core electrons into a general “frozen” potential and therefore the calculation has to only deal with the reactive valence electrons. Due to the applied basis set, periodic boundary conditions were imposed. All the calculations were performed with Quantum-Espresso code.83 Pseudopotentials are used to represent the inner core electrons on average to reduce the computational cost.

To model copper dimers, we use the cluster scheme66: one to four and seven atoms and then the same clusters subsequently surrounded by one helium atom in the ‘T’ and ‘L’ configurations to evaluate the degree of van der Waals binding. The box size of a0 = 15.88 Å, was large enough to keep the cluster isolated from the boundaries and avoid the interference from the periodic boundaries. We used a kinetic energy cutoff of 40 Ry, which reduces the basis set to a computationally affordable size84, and spin polarization in all the calculations through the Quantum Espresso code.83 The Quantum Espresso code is an open computation calculation method program that utilizes DFT to model on a nanoscale.82

Results

For this body of work the equilibrium bond length, vibrational frequencies and binding energies were calculated. The vibrational frequency was determined from the force constant and the following equation

46 (33) ,

where c is the speed of light, μ is the reduced mass and k is the force constant (∝ ).

Helium binding energy was defined as the energy required to detach He from the copper cluster

(34) Binding Energy = E(Cux-He) - [E(He) + E(Cux)], where E(Cux-He) is the energy of the copper cluster – He interaction, E(He) is the energy of the He atom, and E(Cux) is the energy of the cluster alone.

-1 Table (7): Vibrational Frequencies of Cu2 (cm ) Dimer GGA LDA Copper (nonspin 275 266.5 polarized) Computationally 251 calculated Literature Value59

As shown in Tables 7 and 8, DFT provides reasonable accuracy for Cu2 near the equilibrium geometry. The DFT calculated geometries for the trimer and tetramer are shown in Table 8. The trimer exhibits similar Cu – Cu bond lengths as in the dimer but the tetramer bond lengths are less consistent and appear to depend strongly on the DFT model used. Because of this model dependency, the CC method would be a more reliable method to describe copper clusters but is also computationally much more expensive.

47 Table (8): DFT geometry optimized Copper – Copper Bond lengths (Å) GGA LDA Dimer Cu – Cu: 2.22 Cu – Cu: 2.15 Dimer (Experimental 2.22 Literature Value)54 Trimer

Tetramer

Heptamer Cu – Cu: 2.42 Cu – Cu: 2.34 Bulk Metal Bond Length 2.35 (Experimental Literature Value)85

2.28 2.22 2.28

2.49 a. b.

2.42

2.37

2.26

c. d.

Figure (13): Optimized copper cluster geometries: a. Copper dimer (2.22 Å), b. Copper trimer (2.28 and 2.49 Å), c. Copper tetramer (2.37 and 2.26 Å) and d. Copper heptamer (2.42 Å).

48 Table (9): Binding Energy He – Cu2 (meV) Configuration GGA LDA CC (Copper 2) Linear 300 150 1.5 Perpendicular 30 20 0.95 Literature Value 1.05 (Experimental)85

Figures 13 and 14 are visualizations of all Quantum Espresso calculations. Figure 13 is the visual representation of the copper dimers (Cu2), trimers (Cu3), tetramers (Cu4) and heptamers (Cu7). Note that in all molecules, except Cu7, the bond length is found to be approximately 2.2 – 2.4 Å.

3.99

Figure (14): Visualization of Copper – Helium interaction. The bond length between one copper (blue) atom and one helium (white) atom is 4.0 Å.

Figure 14 is the representation of the Cu – He interaction. Note that the bond length between the blue copper atom and the white helium atom is 3.99 Å. For Figure

10, the interactions between the two different conformations of Cu2 – He are shown. The first conformation shown is the T-shaped conformation, where the helium atom is perpendicular to the copper dimer. The second conformation is the L-shaped conformation, where the helium atom is in line with the copper dimer.

49 Discussion

The DFT calculations showed that Cu – Cu bonds are relatively close to the experimental Cu – Cu bond length of 2.2 Å.59 Although the CC method is a more accurate computation method, and DFT does not handle van der Waals interactions well,

DFT appeared to perform reasonably well solely by coincidence. This is consistent with the existing literature86-87 comparing DFT calculations for systems with van der Waals interactions. For the Cu2 – He cluster, DFT was only able to predict the relative energetics for the L and T states but the actual binding energies are off by several orders of magnitude, for example, the binding energy for the linear configuration of Cu2 - He in the DFT method is much higher (0.30 eV vs. 0.0015 eV) than the CC method binding energy (see Table 9). Based on this we expect the correct geometries but incorrect binding energies for larger Cux – He interactions. It is apparent that the applied DFT method is completely unable to deal with van der Waals binding interactions due to the lack of consistent data in regards to binding energies and bond lengths. Although DFT would have been a much faster calculation to use rather than CC, it unfortunately, does not work when van der Waals interactions are present unless other DFT methods are used that can handle dispersion forces.

50 References

1. Atkins, P.; Ronald, F., Molecular Quantum Mechanics. Oxford University Press: United States, 2005.

2. Jensen, F., Introduction to Computational Chemistry. 1st edition ed.; John Wiley and Sons Ltd.: England, 1999.

3. Hartree, D. R., Mathematical Proceedings of the Cambridge Philosophical Society 1927, 24 (1), 111-132.

4. Fock, V. A.; Belinfante, F. J., Physics Today 1980, 33 (8).

5. Helgaker, T.; Jorgensen, P.; Olsen, J., Molecular Electronic-Structure Theory. John Wiley and Sons Ltd.: England, 2000.

6. Richards, W. G.; Horsley, J. A., Ab Initio Molecular Calculations for Chemists. Clarendon Press: Oxford, 1970.

7. Kowalski, K.; Hammond, J. R.; de Jong, W. A.; Sadlej, A. J., J. Chem. Phys. 2008, 129 (22).

8. Hinze, J., J. of Chem. Phys. 1973, 59 (12).

9. Yamaguchi, K., International Journal of Quantum Chemistry 1980, 18 (S14).

10. Avery, J., Creation and Annihilation Operators. McGraw-Hill: New York, 1976.

11. Rohlfing, E. A.; Valentini, J. J., J. Chem. Phys. 1986, 84 (12).

12. Bondybey, V. E.; English, J. H., J. Phys. Chem 1983, 87, 4647 - 4650.

13. Cargnoni, F.; Kus, T.; Mella, M.; Bartlett, R., J. Chem Phys. 2008, 129 (204307), 1-12.

14. Dion, M.; Rydberg, H.; Schroder, E.; Langreth, D. C.; Lundqvist, B. I., Phys. Rev. Lett. 2004, 92.

15. Eloranta, J.; Apkarian, V. A., J. Chem Phys. 2002, 117 (22).

16. Klimes, J.; Bowler, D. R.; Michaelides, A., J. Phys.: Condens. Matter 2009, 22 (2010), 1-5.

51 17. Fiedler, S. L.; Mateo, D.; Aleksanyan, T.; Eloranta, J., Phys. Rev. B 2012, 86.

18. Krylov, A. I., Annu. Rev. Phys .Chem. 2008, 59, 433-462.

19. Wang, S. Ab initio quantum mechanical applications to molecular systems from triatomics to nucleic acid bases. University of Georgia, USA, 2006.

20. Stanton, J. F.; Bartlett, R. J., J. Chem. Phys 1993, 98, 7029 - 7039.

21. Christoffersen, R. E., Basic Principles and Techniques of Molecular Quantum Mechanics Springer-Verlag: New York, 1989.

22. Gutowski, M.; Chalasinski, G., J. Chem. Phys. 1993, 98 (7).3

23. Boys, S. F.; Bernardi, F., Mol. Phys. 1970, 19 (4), 553-566.

24. Hollas, J. M., Modern Spectroscopy. 4th edition ed.; John Wiley and Sons Ltd.: England, 2004.

25. Johansson, A.; Kollman, P.; Rothenberg, S., Theoret. Chim. Acta 1973, 29, 167- 172.

26. Takatani, T.; Hohenstein, E. G.; Sherrill, C. D., J. Phys. Chem. 2008, 128 (12), 124111-124118.

27. Koch, W.; Holthausen, M. C., A Chemist's Guide to Density Functional Theory. 2nd edition ed.; Wiley: Weinheim, 2001.

28. Kohn, W.; Sham, L. J., Phys. Rev. B. 1965, 15.

29. Clark, S. First Principles Molecular Dynamics. United Kingdom, 1996.

30. Foresman, J. B.; Frisch, A., Exploring Chemistry with Electronic Structure Methods. Second Edition ed.; Gaussian, Inc.: Pittsburgh, PA, 1996.

31. Mehta, S.; Price, G. D.; Alfe, D., J Chem. Phys. 2006, 125 (19).

32. Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Pople, J. A., Journal of Chemical Physics 1997, 106, 1063 - 1079.

33. Richards, W. G.; Cooper, D. L., Ab initio Molecular Orbital Calculations for Chemists. 2nd edition ed.; Oxford Press: New York, 1983.

34. Luo, Y., Comprehensive handbook of chemical bond energies. CRC PRess: 2010.

52 35. Barysz, M.; Sadlej, A., J. Theo. Chem. 2001, 573, 181 - 200.

36. Bergman, G., Phys. Rev. Lett. 1981, 48 (15), 1046 - 1049.

37. Blume, M.; Watson, R. E., Proc. R. Soc. Lond. A 1962, 270, 127 - 143.

38. Blume, M.; Watson, R. E., Proc. R. Soc. Lond. A 1963, 271, 565 - 578.

39. Wolf, A.; Reiher, M.; Hess, B. A., J. of Chem. Phys. 2002, 117 (20).

40. Nakajima, T.; Hirao, K., J. of Chem. Phys. 2003, 119 (8).

41. van Lenthe, E.; Snijders, J. G.; Baerends, E. J., J. Chem. Phys. 1996, 105 (15), 6505 - 6516.

42. Winkler, R., Spin-orbit coupling effects in two-dimensional electron and hole systems. Springer Verlag: 2003; Vol. 191.

43. Pacios, L. F.; Christiansen, P. A., Ab initio relativistic effective potentials with spin-orbit operators. I. Li through Ar. In J. Chem. Phys., 1985; Vol. 82, pp 2664 - 2671.

44. Bone, R. G. A.; Bader, R. F. W., J. Phys. Chem. 1996, 100 (26).

45. Menconi, G.; Tozer, D. J., Chem Phys. Lett. 360 2002, 360 (1-2,3), 38-46.

46. Chelikowsky, J. R.; Troullier, N.; Wu, K.; Saad, Y., Phys. Rev. B 1994, 50 (16).

47. Zhang, Y.; Pan, W.; Yang, W., J. Chem Phys. 1997, 107 (19).

48. Becke, A. D., J. Chem Phys. 1982, 76.

49. Jaksic, V.; Pillet, C., Ann. Inst. H. Poincare 1995, 62.

50. Autschbach, J.; Ziegler, T., J. Chem. Phys. 2000, 113 (21), 9410 - 9418.

51. Toennies, J. P.; Vilesov, A. F.; Rev. Phys. Chem. 1998, 49 (1)

52. Hui, Q.; Takami, M.; J. Low Temp. Phys. 2000, 119 (393)

53. Kinoshita, T.; Fukuda, K.; Takahashi, Y.; Phys. Rev. A. 1995, 52 (2707)

54. Hammond, R. L; Lester, W. A.; Reynolds, P. J., Monte Carlo Methods in Ab Initio Quantum Chemistry. World Scientific, 1994

53 55. Dalfovo, F.; Lastri, A.; Pricaupenko, L.; Stringari, S.; Triener, J.; Phys. Rev. B 1995, 52 (1193)

56. Knickelbein, M. B, Chem. Phys. Lett. 1992, 192 (1)

57. Powers, D. E.; Hansen, S. G.; Geusic, M. E.; Pulu, A. C.; Hopkins, J. B.; Dietz, T. G.; Duncan, M. A.; Langridge-Smith, P. R. R.; Smalley, R. E., J. Phys. Chem. 1982, 86 (14)

58. Pettiette, C. L.; Yang, S. H.; Craycraft, M. J.; Conceicao, J.; Laaksonen, R. T.; Cheshnovsky, O.; Smalley, R. E., J. Chem. Phys. 1988, 88.

59. Vehmanen, E.; Ghazarian, V.; Sams, C.; Khachatryan, I.; Eloranta, J.; Apkarian, V., Journal of Physical Chem A 2011, 115 (25), 7077-7088.

60. Jimenez-Hoyos, C. A.; Henderson, T. M.; Scuseria, G. E., Journal of Chemical Theory and Computation 2011, 7 (9).

61. Bhattacharyya, A.; Papenbrock, T., Phys. Rev. A 2006, 74 (4).

62. Page, R. H.; Gudeman, C. S., J. Chem. Phys. 1991, 94 (39).

63. Lochet, J., J. Phys. B 1978, 11 (L5).

64. Ram, R. S.; Jarman, C. N.; Bernath, P. F., J. Mol. Spectr. 1992, 156 (468).

65. McCaffrey, J. G.; Bennet, R. R.; Morse, M. D.; Breckenridge, W. H., J. Chem. Phys. 1989, 91 (92). 66. Morse, M. D., Chem. Rev. 1986, 86 (1049).

67. Ancilotto, F.; Barranco, M.; Hernandez, E. S.; Pi, M., J Low Temp Phys 2009, 157.

68. Royston, J. P., Appl. Statist. 1982, 31 (2).

69. Shapiro, S. S.; Wilk, M. B., Biometrika 1965, 52 (3/4).

70. Gole, J. L.; English, J. H.; Bondybey, V. E., J. Phys. Chem 1982, 86, 2560- 2563.

71. Cargnoni, F.; Kus, T.; Mella, M.; Bartlett, R. J., Journal of Chemical Physics 2008, 129 (204307), 1-12.

72. Carsky, P.; Urban, M., Ab Initio Calculations: Methods and Applications in Chemistry. Springer-Verlag: New York, 1980; Vol. 16.

54 73. Fiolhais, C.; Noguiera, F.; Marques, M., A Primer in Density Functional Theory. Springer: Berlin, 2003.

74. Aiken III, J. D.; Finke, R. G., Journal of Molecular Catalysis A: Chemical 1999, 145 (1-2).

75. Boudart, M., Journal of Molecular Catalysis 1985, 30 (1-2).

76. Geusic, M. E.; Morse, M. D.; Smalley, R. E., J. of Chem. Phys. 1985, 82 (1).

77. Morse, M. D.; Geusic, M. E.; Heath, J. R.; Smalley, R. E., J. of Chem. Phys. 1985, 83 (5).

78. Stark, J.; Park, D. G.; Lagadic, I.; Klabunde, K. J., Chemistry of Materials 1996, 8 (8).

79. Xu, Z.; Xiao, F. S.; Purnell, S. K.; Alexeev, O.; Kawi, S.; Deutsch, S. E.; Gates, B. C., Nature 1994, 372.

80. Yanagisawa, S.; Tsuneda, T.; Hirao, K., J. of Chem. Phys. 1999, 112 (2).

81. Perdew, J. P.; Burke, K.; Ernzerhof, M., Phys. Rev. Lett. 1996, 77.

82. Kresse, G.; Furthmuller, J., Computational Materials Science 1996, 6 (1).

83. Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Dal Corso, A.; de Gironcoli, S.; Fabris, S.; Fratesi, G.; Gebauer, R.; Gerstmann, U.; Gougoussis, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.; Marzari, L.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.; Scandolo, S.; Sclauzero, G.; Seitsonen, A. P.; Smogunov, A.; Umari, P.; Wentzcovitch, R. M., J. Phys.: condens. Matter 2009, 21 (39).

84. Car, R.; Parrinelloa, M., Phys. Rev. Lett. 1985, 55.

85. Pauling, L.; J. Am. Chem. Soc. 1947, 69

86. Delley, B.; Ellis, D. E.; Freeman, A. J., Phys. Rev. B. 1983, 27 (4).

87. Sun, Y.; Kim, Y.; Lee, K.; Zhang, S. B., J. Chem. Phys. 2008, 129

88. Andersson, Y.; Langreth, D. C.; Lundqvist, B. I, Phys. Rev. Lett. 1996, 76 (1)

89. Ahokas, J.; Kiljunen, T.; Eloranta, J.; Kunttu, H., J. of Chem. Phys. 2000, 112 (5). 90. Boettger, J. C., Phys. Rev. B. 2000, 62 (12), 7809 - 7815.

55

91. Bohm, D., Quantum Theory. Dover Publications, Inc.: New York, 1979.

79. Bondybey, V. E.; English, J. H., Journal of Physical Chemistry 1983, 87, 4647- 4650.

80. Born, M., Atomic Physics. Eighth Edition ed.; Dover Publications, Inc: New York, 1969.

81. Brockmann, R.; Weise, W., Phys. Rev. C. 1977, 16 (3), 1282 - 1284.

82. Da Silva, J. L. F.; Stampfl, C.; Scheffler, M., Phys. Rev. Ltrs 2003, 90 (066104), 1-4.

83. Ermler, W. C.; Lee, Y. S.; Christiansen, P. A.; Pitzer, K. S., Chem. Phys. Lett. 1981, 81 (1), 70 - 74.

84. Ernzerhof, M.; Scuseria, G. E., J. of Chem. Phys. 1998, 110 (11).

85. Ferrario, M.; Ciccotti, G.; Binder, K., Computer Simulations in Condensed Matter: From Materials to Chemical Biology. Springer: New York, 2006; Vol. 1.

86. Gropen, O., J. Theo. Chem. 2005, 727, 159 - 164.

87. Heb, B. A., Ber. Bunsenges. Phys. Chem. 1997, 101, 1 - 10.

88. Hinze, J., J. of Chem. Phys. 1973, 59 (12).

89. Hirata, S.; Yanai, T.; de Jong, W. A.; Nakajima, T.; Hirao, K., J. Chem. Phys. 2004, 120, 3297 - 3309.

90. House, J. E., Fundamentals of Quantum Chemistry. 2nd edition ed.; Elsevier Academic Press: San Diego, CA, 2004.

91. Inglis, D. R., Letter to the Editor: Spin-Orbit Coupling in Nuclei. Letters to the Editor ed.; Pittsburgh, Pennsylvania, 1936; p 783.

92. Koch, W.; Holthausen, M. C., A Chemist's Guide to Density Functional Theory. Wiley: 2001.

93. Korona, T.; Werner, H.-J., J. Chem. Phys. 2004, 118 (7), 3006-3019.

94. Kucharski, S. A.; Wloch, M.; Musial, M.; Bartlett, R. J., J. Chem. Phys. 2001, 115, 8263 - 8266.

56 95. Labanowski, J. K. Simplified Introduction to Ab Initio Basis Sets. Terms and Notation. (accessed 1/10/13).

96. Liberles, A., Introduction to Molecular-Orbital Theory. Holt, Rinehart adn Winston, Inc.: USA, 1966.

97. Lowe, J. P., Quantum Chemistry. Academic Press: New York, 1978.

98. Marx, D.; Hutter, J., Modern Methods and Algorithms of Quantum Chemistry. In Ab initio molecular dynamics: Theory and Implemtation [Online] John von Neumann Institute for Computing: 2000; pp. 1-77.

99. Matveev, A.; Rosch, N., J. Chem. Phys. 2003, 118, 3997 - 4012.

100. Musolino, V.; Selloni, A.; Car, R., Journal of Physical Chemistry 1998, 108 (12), 5044 - 5054.

101. Musolino, V.; Selloni, A.; Car, R., J. Chem Phys. 1998, 108, 5044-5054.

102. Perez-Jorda, J.; Becke, A. D., Chem. Phys. Lett. 1995, 233 (1-2).

103. Quinn, C. M., Computational Quantum Chemistry. Academic Press: San Diego, CA, 2002.

104. Ratner, M. A.; Schatz, G. C., Introduction to Quantum Mechanics in Chemistry. Prentice Hall: New Jersey, 2001.

105. Reiher, M., Theor. Chem. Acc. 2006, 116, 241 - 252.

106. Rohlfing, E. A.; Valentini, J. J., J. Chem. Phys. 1986, 84 (6560).

107. Roos, B. O.; Lindh, R.; Malmqvist, P.-A.; Veryazov, V.; Widmark, P.-O., J. Phys. Chem. 2004, 108, 2851 - 2858.

108. Sams, C.; Eloranta, J., Ab Initio Calculation of Copper Dimer Excited States. 2011 (powerpoint presentation).

109. Sams, C.; Satta, A.; Zipoli, F.; Selloni, A.; Eloranta, J.; Kioussis, N., Conformations of Solvated Divalent Metal Clusters. 2009 (powerpoint presentation).

110. Silbey, R. J.; Alberty, R. A.; Bawendi, M. G., Physical Chemistry. 4th Edition ed.; John Wiley and Sons Ltd, 2005.

57

Appendix A

Article Reprinted with Permission from Injection of Atoms and Molecules in a Superfluid Helium Fountain: Cu and Cu2Hen (n = 1, ..., ∞) Esa Vehmanen, Vahan Ghazarian, Courtney Sams, Isahak Khachatryan, Jussi Eloranta, and V. A. Apkarian. The Journal of Physical Chemistry A 2011115 (25), 7077-7088. Copyright 2011 American Chemical Society

58 Injection of Atoms and Molecules in a Superfluid

Helium Fountain: Cu and Cu2Hen (n = 1...∞)

Esa Vehmanen,†,‡,¶ Vahan Ghazarian,‡,§ Courtney Sams,¶ Isahak Khachatryan,‡

¶ ,‡ Jussi Eloranta, and V. A. Apkarian∗

Nanoscience Center, Department of Chemistry, P.O. Box 35, FIN–40014 University of Jyväskylä, Finland, Department of Chemistry, University of California, Irvine, CA 92697, Department of

Chemistry and Biochemistry, California State University at Northridge, 18111 Nordhoff St., Northridge, CA 91330, and Department of Chemistry, East Los Angeles College, Monterey Park, CA 91754

E-mail: [email protected]

∗To whom correspondence should be addressed †Nanoscience Center, Department of Chemistry, P.O. Box 35, FIN–40014 University of Jyväskylä, Finland ‡Department of Chemistry, University of California, Irvine,CA92697 ¶Department of Chemistry and Biochemistry, California StateUniversityatNorthridge,18111NordhoffSt., Northridge, CA 91330 §Department of Chemistry, East Los Angeles College, MontereyPark,CA91754

1 Abstract

We introduce an experimental platform designed around a thermo-mechanical helium foun-

tain, which is aimed at investigating spectroscopy and dynamics of atoms and molecules in the

superfluid, and at its vapor interface. Laser ablation of copper, efficient cooling and transport

of Cu and Cu2 through helium vapor (1.5 K < T < 20 K), formation of linear and T-shaped

Cu2-He complexes, and their continuous evolution into large Cu2-Hen clusters and droplets, are among the processes that are illustrated. Reflection is the dominant quantum scattering

channel of translationally cold copper atoms (T = 1.7 K) at the fountain interface. Cu2 dimers mainly travel through the fountain unimpeded. However, the conditions of fountain flow and

transport of molecules can be controlled to demonstrate injection, and in particular injection

into a non-divergent columnar fountain with a plug velocity of ca. 1 m/s. The experimental

observables are interpreted with the aid of bosonic density functional theory calculations and

ab initio interaction potentials.

Introduction

Spectroscopic studies of molecules in superfluid helium is anadvancedsubject.1,2 While driven by diverse motivations, the investigations inevitably probe implications of superfluidity on molecular

scales.3 Of the methods used, the cluster pick-up scheme has proven to be most effective.4 And al- though a tremendous amount has been learned, the method has several limitations. The supersonic expansion fixes the thermodynamic P,T state of clusters and eliminates temperature as a variable { } –aparameterthatplaysakeyroleincontrollingtheproperties of He-II.5,6 Moreover, at the char- acteristic temperature of clusters (0.37 K),7 rotons are not thermally excited and the phonon cut-off

wavelength (ca. 5nm)impliesthattheycanonlyexistinclusterswithmorethan 103 atoms. This leaves the droplet surface and surface capillary waves (ripplons) as the principal source and sink of

excitations, along with excitation that may be imparted during the pick-up process. It would be de- sirable to eliminate these constraints. To this end, we have considered several different approaches to inject molecules in bulk-like liquid. Of these, injectionoflaser-ablatedmoleculesinlargeliquid

2 droplets produced by pulsed supersonic expansion,8 has enabled extensions of the cluster pick-up

approach.9–12 Historically, the first spectroscopic observations of molecules in He-II were real- ized in the bulk liquid.13,14 There have been significant efforts to introduce both neutrals15–17 and ions18 in static liquid and solid helium; including the recent spectroscopic study of copper dimers in liquid and solid helium.19 The mundane consideration that all molecular species freezeand phase separate at the relevant temperatures dictates transient conditions for measurements in the

bulk, and methods such as laser ablation in the bulk, commonlyresultinill-definedthermodynamic states. A flowing liquid, with a continuously refreshed volume could in principle overcome this

limitation. It is in this context that we consider the injection of molecules in a thermo-mechanical superfluid fountain,5,6,20 which is the subject of the present report.

We describe our experimental effort to inject laser ablated copper atoms and dimers in a helium fountain. Under the experimental conditions we employ, translational energies thermalized to 1.5 ! Kandwithvelocitiesrelativetothemovingsurfaceof 1m/s,theinjectionofatomsandmolecules ! into a flowing superfluid column raises fundamental questionsregardingscatteringdynamicsatthe ideal liquid-gas interface. For weakly interacting systems, such as Cu-He, quantum evaporation

and condensation can be expected to control the interfacial scattering dynamics.21–23 Entrainment of molecules in the flow requires momentum transfer, for whichthereisnotanobviousmechanism

in the volume of the fountain, below the critical velocity of He-II ( 20 m/s). Barring vorticity, in ! the absence of viscous drag on the translational motion of molecules, it is not clear whether they can be carried by the flow or whether they would pass through a superfluid fountain unimpeded. Once

again, one can expect excitations at the liquid-gas interface to play a major role in the scattering processes, now at the gas-fountain interface at the relatively high temperature of 1.5 K. Key to the ! successful implementation of these experiments is characterization of the transport and cooling of plasma ablated atoms and molecules in the cold vapor of liquidhelium.Laserablationofcopper,

which has been extensively investigated in different contexts,24–26 proves particularly valuable for

27–33 visualizing the relevant transport dynamics. The characterized visible spectrum of Cu2 makes it a convenient probe of the essential dynamical processes. Rapid cooling and neutralization of

3 the plasma, dimer formation through three-body collisions and on droplets, rotational cooling and

freeze-out of the vibrational population, subsequent pick-up of helium atoms and cluster growth are among the processes that can be tracked as a function of time and distance the molecules

travel through cold helium. We outline these processes, withthemainaimofreportingproofof

principle: the injection of atomic Cu and molecular Cu2 in the fountain, and the unique aspects our experimental platform offers for spectroscopic and dynamical investigations. The latter are highlighted by the accessible wealth of spectroscopic information regarding Cu2 and Cu2-Hen (n = 1,...,∞)clusters,whichwepresentalongwiththeirtheoreticalanalysis.

Methods

Experimental

The experimental setup is shown in Figure 1a. We use a triple walled liquid helium Dewar, with an inner diameter of 7.5 cm, and with optical access provided through five sets of triple windows (modified Oxford Variox). The liquid level is controlled withaneedlevalve,thetemperaturein- side the cryostat is measured with a rhodium iron sensor (Oxford model T1-103), the pressure is monitored with a capacitance manometer (MKS Baratron). The cryostat is pumped with a high speed, 300 cfm, roots vacuum pump (Alcatel Adixen), which under typical operation conditions allows the reduction of the pumped helium temperature to 1.5 K. The pumping speed is regu- lated with a feedback controlled throttle valve (MKS model 252) to maintain the vapor at constant

6 pressure. The vacuum shroud is maintained at 10− torr using a separate turbo molecular pump

(Pfeiffer model TPH 062; 56 L/s of N2). The fountain consists of a glass tube packed with 0.5 - 2 µmrouge(ferricoxideparticles)anda30Ω resistive heater. The tube is drawn into a capillary with an inner diameter of 200 µm, to serve as the fountain spout. The flow rate of the fountain is controlled with the heater, relying on the thermo-mechanical effect to generate a differential pres-

sure ΔP = ΔS(T)ΔT,withentropydensitycarriedbythenormalfractionacrossthe temperature gradient. The flow velocity at the spout is deduced by the height (h)ofthefountain,assuming

4 deceleration strictly due to the force of gravity: v = √2gh.Thefountaincanbeoperatedinsev-

eral flow modes (see Figure 1b). A non-divergent column flow is obtained with heights exceeding 1cm,atflowvelocitiesabove 0.5 m/s. By reducing the pressure to only overcome surface ! ! tension, stationary free-standing liquid profiles (“blobs”) can be generated, with residual surface flow ensuring a refreshed interface. Intermediate between these limits, is the misty-top operation, the characteristics of which are determined by the speed at which the vapor is pumped. In all but the last case, a sharp liquid-gas interface is obtained, as evidenced by the light diffraction that generates the contrast in the images of Figure 1b.

Two different arrangements have been used to rotate the laserablationtarget,tomaintainafresh ablation surface. In the initial studies, the target consisted of an oxygen-free high conductivity

(OFHC) copper disk, mounted on a motor with a horizontal shaft.Thedisk-to-fountaindistanceis adjusted, with typical operating conditions correspondingtoaseparationof1-2mm.Fluorescence imaging of the ablation products (i.e., Cu atoms and Cu2)isusedforvisualizinggasdynamicspost ablation. Analysis of the flow indicates convective drift duetoathermalgradientbetweenthe motor-mounted disk and the fountain. Replacement of the target with a rod suspended from a vertically mounted motor at the top of the cryostat, with the rod partially submerged in the liquid reservoir (see Figure 1a), effectively eliminated the convective gas flow.

A XeCl excimer laser, operating at 308 nm and a repetition rate of 20 Hz, is used to ablate the copper (Lambda Physik EMG 101 MSC; 80 mJ/pulse). A time-delayed Nd-YAG pumped dye laser is used for the spectroscopic measurements (Continuum Surelite, Lambda Physik FL3002).

1 The spectral width of the free running dye laser is 0.2 cm− .Thelinenarrowsdownto0.04 1 cm− with the insertion of an intracavity etalon. An optogalvaniccell(Sirah)isusedforabsolute wavelength calibration. The ablation and dye laser beams arefocusedwithseparatequartzlenses and combined using a 50/50 beam splitter. An intensified charge coupled device (ICCD; Princeton

PI-MAX; 2 ns minimum gate width) camera equipped with a macro lens, is used to obtain time- gated spatial images of the fluorescence. Alternatively, thelaser-inducedfluorescenceiscollected with two plano-convex lenses, dispersed in a 1/4-meter monochromator, and detected using an

5 a)

b)

Figure 1: a) Schematic overview of the experimental setup. A triple-walled liquid helium cryostat with optical access (windows shown in gray) is used. The fountain flow is controlled through resistive heating, the ablation target is a copper rod partially submerged in the liquid, and rotated through the vertically mounted motor. The excimer laser is used for ablation, and the dye laser is used for LIF measurements. The block diagram shows the optical arrangement. An intensified CCD array, combined with a macro lens allows imaging with time resolution of 2 ns (minimum gate width of ICCD). When employed behind the monochromator (M), the ICCD provides time- gated, spectrally-resolved emission spectra. The timing between excimer, dye-laser, and detection gate is adjustable. b) Distinct modes of operation of the thermo-mechanical fountain (small blob, large blob, misty mode and non-divergent column flow). The fountain diameter is 200 microns.

6 ICCD array to record time-gated emission spectra. Timing between the two lasers and the ICCD

gate is provided by software controlled delay generators (SRS model DG 535). The system was controlled by a computer, using the libmeas package.34

Theory

We compute the copper dimer (within Abelian point group D2h;coreorbitals1s,2s,2p,3s,3p on both atoms) ground and several low lying excited states (e.g., X, A, B, C states) around the equi- librium geometry using the closed-shell equation-of-motion (EOM) coupled clusters (CC) method

with single and double excitations (SD), based on a single restricted Hartree-Fock (RHF) refer- ence.35 Due to the single reference nature of this approach, the potentials are only reliable near the

equilibrium geometry. Single reference based methods give incorrect molecular dissociation, as

36 evidenced by the rapid increase of T1 diagnostic value far away from the equilibrium geometry. We include scalar relativistic effects for Cu through the second order Douglas-Kroll-Hess (DKH) Hamiltonian and apply a compatible correlation consistent basis set, aug-cc-pwCVTZ.37–40 The closed-shell EOM-CCSD and the following CCSD(T) calculations were carried out with the MOL-

PRO code.41 For excited states, the effect of spin-orbit coupling was also considered as this may mix the electronic singlet and triplet states. Our main interest was in the characterization of the A state, to which the transition from the ground state is spin forbidden. Spin-orbit coupling mixes

3 1 + 42 the A( Πu) and B( Σu )statesasfollows:

3 E( Πu) ξ/2 √6ξ/2 − (1)  √ 1 +   6ξ/2 E( Σu )

where ξ is the spin-orbit coupling of Cu atom (0.1 eV). The relative energies of the A and B states were calculated by using the DKH-EOM-CCSD/aug-cc-pwCVTZ method, based on unrestricted

Hartree-Fock (UHF) reference. This single point calculation was carried out with the Gaussian 09 code.43

7 The ground state Cu-He and Cu2-He calculations employed the standard CCSD(T) method (core orbitals 1s,2s,2p,3s,3p on both Cu atoms), where in addition to single and double exci- tations, iterative triples were included.44,45 Given the very weak van der Waals-type interaction, it was necessary to employ a large basis set, aug-cc-pV5Z (AV5Z),40,46 to treat electron correla- tion to high accuracy, and to apply the basis set superposition error (BSSE) correction to remove

47 deficiencies from using an incomplete basis set. The calculation was carried out for He-Cu2-He supermolecule within D2h symmetry with the resulting energies divided by two to give the effective

Cu2-He interaction. Note that the He-He interaction at the distances of relevance have negligible contribution to energy. The applied bosonic density functional theory (DFT) model todescribesuperfluidheliumand the numerical implementation has been described previously. 48–50 The ground state solutions for the liquid interacting with Cu or Cu2 at 0 K temperature were obtained by the imaginary time propagation method with a 40 fs time step and a 256 256 256 spatial grid with a grid step of 0.4 × × Bohr. The droplet energies were evaluated by numerically integrating the resulting energy density.

Results

Copper atoms

Time-gated fluorescence images of the ablation plume are shown in Figure 2a. Each image is obtained as 40 averages, with a gate width of 100 ns, at the indicated gate delay relative to the

ablation laser pulse. The expansion is against cold helium vapor (P =10torr,T =1.7K).Thefront

of the mushroom shaped plume at 150 ns is at 400 µmfromtheablationtarget;itevolvesintoa crescent at t 200 ns, characteristic of detached shock front with a bow velocity of 103 m/s; post t ! 300 ns, the directed motion is arrested and the plume evolves into isotropic expansion. To follow ! the history of the expansion post 500 ns (limited by the radiative lifetime of the metastable Cu

32P0 32D emission; τ = 500 ns),51 we resort to laser induced fluorescence (LIF) imaging. 3/2 → 5/2 rad We use a doubled dye laser tuned to the atomic 42S 32P0 transition at 324.75 nm, while 1/2 → 3/2

8 monitoring the 32P0 32D transition at 510.55 nm.51 LIF images of the Cu atom recorded 3/2 → 5/2 at time delays as long as 1.5 ms are shown in Figure 2b. In this duration, the principal change in the distribution is the reduction of the atom number density in the probed volume. This can be discerned from the reduction in the vertical width of the fluorescence, which is due to fluorescence re-absorption and radiation trapping in the dense Cu gas. The effect allows the estimation of the

13 3 52 atom number density as 10 cm− as the cloud reaches the fountain location at t = 150 µs. Note, the atom distribution appears to stagnate between target and fountain during the 1mstime ! of observation. This would be expected for a thermal velocitydistributionundergoingdiffusive

expansion. At T = 2K,adiffusionconstantofD = 2 10 6 m2/s is to be expected, therefore an × − isotropic expansion of √Dt 50 µmin1ms,somewhatlargerthantheobservedspreadingof 30 ! ! µm/ms. In effect, the atomic translational distribution is thermalized on the timescale of 1 ms. The positions of the target surface and the fountain are marked in Figure 2b, at t = 750 µs the atom cloud extends past the fountain. The effect of the fountain on the spatial distribution of

atoms, along the path of the dye laser, is obtained by recording LIF images with fountain on and

off. Difference plots of the spatial LIF distribution, I I ,recordedatt = 750 µsand1.5ms, on − off are shown in Figure 3. Upon turning the fountain on, a total deficit in LIF intensity of 20 - 40% develops behind it. The depletion reaches its maximum depth 2diametersbehindthefountain ! –ashadowcastbyanopaquefountain.Precedingthelosscurve, there is a clear pile-up of LIF in front of the fountain, in the 1.5 ms data (Figure 3b). The effect is much smaller in the 750

µsdata(Figure3a).Inthesedirectline-of-sightmeasurements (where the dye laser crosses the fountain at 90◦,seegeometryinFigure1a),pile-upofatomsatthefrontofthe fountain implies reflection at the gas-liquid interface. Note that the lost atom flux, which significantly exceeds the reflection pile-up, does not have a unique interpretation. Deflection of the flux out of the plane of observation (defined by the intersection of the fountain and the dye laser beam), would explain the shadow cast behind the fountain. Alternatively, entrapmentinthebulkorsurfaceofthefountain would lead to the same depletion pattern given that the fluorescence of wet atoms is quenched. We have searched but failed to find spectroscopic signatures of entrained atoms. Independent of this,

9 Figure 2: Spatial evolution of the ablation plume in cold helium gas (2 K temperature and 10 Torr pressure) as a function of the delay between the excimer and the doubled dye lasers (emission intensity is given in arbitrary units by contour colors with red and blue indicating high and low intensity, respectively). a) Time-gated fluorescence images recorded by monitoring emission over the Cu(32P0 32D )transitionat510.55nm.Thelocationofthecoppertargetisindicated 3/2 → 5/2 by a brown vertical line and the fountain was turned off (magenta line shown only for reference). 2 2 b) Fluorescence images obtained by re-excitation of Cu atoms over the 4 S1/2 3 P3/2 transition at 324.75 nm while monitoring the same emission as in (a). The location of the→ Cu target and the fountain in b) are indicated by brown and magenta vertical lines, respectively. the clear identification of reflection at normal incidence necessarily implies that deflection must contribute to the observed depletion of LIF. The robust conclusion is the observation of efficient reflection of Cu atoms when their translation is thermalized to 1.7 K in 1.5 ms (see Figure 3b). ! We should point out that the observed reflection pile-up at thefrontofthefountainalsoestablishes that aspiration (Venturi effect) does not play a significant role in determining the gas density profile at the liquid-gas interface.

10 Figure 3: The differential spatial distribution of Cu atom fluorescence obtained as Ion Ioff (av- eraged along the axis parallel to the copper target surface).Thegrayverticalbandrepresents− the location of the fountain. The dip in the distribution is the shadow cast by the opacity of the fountain (sum of inelastic and elastic scattering), while the positive peak in front of the fountain represents reflection at normal incidence. The indicated times are the time delays between the ex- cimer and dye lasers, as in Figure 2b. At 1.5 ms, the Cu atom translation is thermalized, the motion is diffusive.

Copper dimers and Cu2-Hen complexes

Copper dimers are formed efficiently in the Knudsen layer, where the ablation plume and the he- lium gas interpenetrate.53 This is established spectroscopically, through two-dimensional excitation- emission spectra as shown, for example, in Figure 4. At the relatively elevated temperatures of T

2.9 - 20 K, over a reservoir of normal helium, the observed spectra can be understood as those ! of bare Cu2.Thereisnotaclearsignatureofcomplexationwithheliumwithin the resolution of

11 the spectral record. The spectrum is dominated by the B(v = 0,...,5) X(v = 0) progression, ← along with sequences built on X(v = 0,...,4). In addition, three weaker vibrational transitions to a state nested in B can be identified. Access to these states leads to emission from the nearest lying vibrational level of the B state. This is directly established in the 2-D spectra, by comparing the emission from each accessed state (vertical axis in Figure 4). The lines can be assigned to C ← 1 X, placing the C state 80 cm− above B. The extracted spectroscopic constants are collectedin Table 1. The B state vibrational constants, Franck-Condon (FC) factors, and electronic origins are nearly identical to those of the bare molecule. Based on the FC factors, we find R R =0.07 | X − C| Å, to be compared with the literature value of 0.04 Å; and R R =0.10Å,tobecompared | X − B| with the literature values of 0.11 Å.29–33,54 The equilibrium bond length in the C state is closer to the X state than in the B state but overall the bond lengths are very similar in all three states.

Table1: ParametersforX,A,a,a’,BandCstatesofCu2 molecule. The electronic origin Te is 1 1 given in cm− ,harmonicandanharmonicfrequenciesωe and ωexe are in cm− ,equilibrium bond length Re in Å, and radiative lifetime in ns.

a b c a b c a b c a b State Te Te Te Re Re Re ωe ωe ωe ωexe ωexe Lifetime 1 + X( Σg ) 0 0 0 2.22 2.22 2.25 275 266.5 251 1.4 1.04 – 3 a( Σu) – 15420 16450 – 2.48 – – 125 – – – – 3 a’ ( Σg)––18710––––––––– 3 1 + d A( Πu/ Σu ) – 20431 20000 – 2.28 – – 192.5 – – 0.35 – 1 + b c B( Σu ) 21767 21758 22260 2.32 2.33 2.38 244 1 246.3 216 2.1 0.1 2.2 20 ,200 1 ± ± b c C( Πu) 21875 21866 22660 2.29 2.26 2.25 218 221.4 240 0.5 1.76 1000 ,1000 a Experimental data obtained in this work. The bond lengths were obtained through the analysis of Frank-Condon factors assuming that the ground state bond length is 2.22 Å. b Data from previous work. 29–33,54 c Theoretical calculations in this work. d Spin-orbit mixing with the B state makes the A state partiallyallowed.

In Figure 5a we show a high-resolution segment of the B(v = 2) X(v = 0) line, recorded using ← an intracavity etalon. The spectrum at 20 K is perfectly reproduced with the known molecular constants, including the unusual isotope dependent electronic origins.33 The fit determines the rotational temperature of the dimer to be the same as the gas temperature. We generally find that at time delays t > 0.1ms,therotationofthedimerisfullythermalizedandthevibrational populations are frozen out. In contrast with the 20 K spectrum, the high-resolution excitation spectrum at 2 K

(Figure 5b) cannot be reproduced under the assumption of bare Cu2. While the rotational envelope of the molecule shows thermalization at Trot 2K,thereissignificantmismatchbetweenobserved ! and predicted lines. There now are additional sharp lines in the observed spectrum, suggestive of

12 Figure 4: Two-dimensional excitation-emission spectrum of Cu2 in dense helium gas at T = 20 K(contourcolorsrepresentthefluorescenceintensityinarbitrary units with red and blue corre- sponding to high and low intensity, respectively). Both B XandC Xvibronictransitionsare labeled in the graph. ← ←

transitions belonging to Cu2-He. Given the sharp resonance, the transition must terminate on a bound long-lived excited state of the complex. Based on the calculated interaction potentials on ground and excited states, we will assign this species to a T-shaped Cu2-He complex. Ablation over pumped superfluid helium at temperatures below2Kgeneratesanewprogres- sion, as seen in the excitation-emission spectrum of Figure 6a. The excitation progression is blue shifted from that of B Xby580cm 1.Theemissionoriginofthenewprogressionappearson ← − adiagonalshiftedfromthatoftheexcitationorigins,indicating that each state emits after losing the same amount of energy. The emitting states are easily assigned to bare Cu (B X) based on 2 → their FC patterns, as shown in Figure 6b. Since the spectrum only develops at T < 2K,itmustbe assigned to a weakly bound complex in the ground state. The displaced origin line clearly shows that the complex undergoes vibrational predissociation in the B state, to emit as bare Cu2. Closer inspection of the trace shown in Figure 6a reveals modulated excitation profiles, which are much broader than those of the bare molecule. They show unresolvedbluetailswithanenvelopethatis

1 characteristically modulated at 30 cm− –structurethatmustbeassociatedwithvibrationsofthe ! 13 Figure 5: High-resolution spectrum of Cu2 corresponding to B(v! = 2) X(v!! = 0) transition at 25 K (panel a) and 2 K (panel b). ←

14 1 complex in the excited state. The excitation profiles also show modulation of 12 cm− on the red ! edges, structure that must arise from the populated vibrational states of the complex in the ground state. Based on either the widths of the excitation profile, or on the relative intensity of the relaxed

emission to that at the origin, a vibrational predissociation time of 1 - 3 ps can be estimated. We in-

fer that Cu2 complexed to helium in a well-defined structure is being seen.Incontrasttothesharp lines observed in the complex identified in the high-resolution spectrum (Figure 5b), the key to the

structural assignment of this complex rests on the well-defined predissociation energy of ca. 600

1 cm− .Thisrepresentstheexcitedstaterepulsiveenergyalongthe Cu2(B)-Hen coordinate at the ground state equilibrium geometry of Cu2(X)-He, which is predicted to occur for linear Cu-Cu-He interactions based on the calculations to be presented below. However, the calculations predict a

1 repulsive energy of only 100 cm− for the linear complex. As such, we refer to this species as

Cu2-Hen(L) to identify that it must contain linear interactions, andthatthecomplexmaycontain more than one helium atom.

Cu2-He dry cluster nucleation and growth

The above already establishes that transport of Cu dimers through cold helium vapor leads to com- plexation with helium. We see variations in the spectra with the conditions of ablation, pressure and temperature of the vapor, and time and distance at which the dimers are intercepted with the excitation laser. A useful method to understand the transport and formation dynamics of various species is through excitation imaging: recording the spatial distribution of fluorescence as a func- tion of excitation wavelength, at a given delay between ablation and re-excitation. Examples of excitation imaging are shown in Figure 7, along with an image of the fountain and the spatial distri- bution of the fluorescence. The background signal without theablationlaserisshowninFigure7a where the tip of the fountain is visible at the bottom. The image in Figure 7b was recorded 1.5 mm above the fountain, with the fountain idling in the blob-mode, whereas a free-standing liquid fountain extends over the full height in Figure 7c. Cuts along the spatial coordinate, Figure 8a, show the evolution of the spectrum as a function of distance traveled. After traveling a distance of

15 a)

v" = 0 Cu -He (L) v’ = 3 210 2 n v" = 0 490 1 0 C - X 480

470

460

450 Emission (nm) 440 2 1 430 B - X v’ = 4 3 2 1 0 0 v" 430 435 440 445 450 455 460 465 Excitation (nm) b) v = 3

440 460 480 500 v = 2

440 460 480 500 v = 1

440 460 480 500 v = 0

440 460 480 500 Emission Wavelength (nm)

Figure 6: a) Two-dimensional excitation-emission spectrum of Cu2 in pumped helium at T = 1.7 K(contourcolorsrepresentthefluorescenceintensityinarbitrary units with green and blue corre- sponding to high and low intensity, respectively). The traceshownisobtainedbyintegratingover the emission coordinate (vertical axis). The diagonal whereexcitationandemissionwavelengths coincide represents the line of origins (unrelaxed emission) of the dimer. The shifted diagonal rep- resents fluorescence after vibrational predissociation of the Cu2-He complex, where Δ is the energy difference between the excited state and the emitting state.b)Exceptfortheshiftinexcitationori- gin (indicated with arrows for the v = 3transition),thefluorescencespectrumfromthecomplex (red) and the dimer (black) are identical. The structure in the emission spectrum is dictated by Franck-Condon factors of the Cu (B X) transition. 2 → 16 d = 1 mm from the ablation target, only the sharp progression of Cu2(X, v = 0) is seen. As the molecules approach the fountain, at d = 2mm,theprogressionstartsdevelopingbluewings(see Figure 8a). The wings grow continuously over the fountain, and well past it. We see the contin- uous growth of a helium cluster as the molecules travel through the cold vapor. The nucleation occurs as the molecules enter a wide cone subtended by the blob; and once initiated, the growth is continuous. The cone can be associated with a region of denser (colder) helium in this pumped geometry. Nevertheless, the growth is homogeneous, gas phase, without direct contact with the liquid. As such, we refer to these clusters as dry Cu2-Hen.Theirspectralsignatureisavibrational progression that builds on the Cu (B, C X) progressions, with a sharp red edge and a long blue 2 ← tail that grows in intensity as a function of distance traveled (Figure 8a).

Cu2-He wet cluster growth

The excitation image in Figure 7c was obtained with the fountain turned on. The spectral cuts as a function of distance from the fountain are shown in Figure8b.Anevolutionsimilartothe case of the dry clusters is seen, with the principal difference in the spectral signature being the disappearance of the sharp red-edge of progressions. The difference is suggestive of not only a size difference between dry and wet clusters, but also of embedding of the molecule. A sharp edge would imply access of zero-phonon lines. Their disappearance implies a solvation structure in which the excited and ground states shift relative to each other along the solvation coordinate.

This characteristic difference suggests that in the dry clusters the molecule remains near, or on, the surface; while in wet clusters, the molecules are embedded. The evolution of these spectra illustrates sensitivity to size, and structure. The more remarkable observation in the excitation images is the absence of any signature of the fountain. Any interaction – reflection, entrainment, condensation by formation of large copper clusters – would beexpectedtoleaveaverticaltracein the image at the fountain position, either due to enhanced or reduced fluorescence from the dimers, or any other new species.

17 Figure 7: Fluorescence images of a) spatial background signal without laser ablation (the fountain spout is visible at the bottom via light scattering); b) excitation image of Cu2 when the fountain is in the blob mode below the ablation/interrogation line that can be seen in (a); c) excitation image of Cu2 when the fountain crosses the ablation/interrogation line.Inb)andc)they-axis corresponds to the excitation wavelength. Note that in the latter two panels the LIF signal was averaged over the spatial y-direction (along the copper target) to enable 2D contour presentation. The x-axis in all panels corresponds to the distance from the ablation target. The contour colors represent the emission intensity in arbitrary units with red and blue corresponding to high and low intensity, respectively.

18 λ λ a) b)

Figure 8: a) Spectral cuts from Figure 7b at the indicated distances from the ablation target. b) Spectral cuts taken from Figure 7c.

Cu2-He(liq) injection and transmission through the fountain

To ensure that we observe the spectrum of the dimer in the liquid phase, without being over- whelmed by the signal from the surrounding clusters, we operate the fountain in its misty mode – where a fine mist is sprayed by the boiling fountainhead (Figure 1). The excitation image is shown in Figure 9a, a cut along the spatial coordinate is shown in Figure 9b, and two spectral cuts, before and after the fountain, are shown in Figure 9c. The very broad spectrum, in which the dimer pro- gression is only apparent as a weak modulation, can be safely assigned to Cu2-He(liq). Note the sharp lines that appear in the spectrum behind the fountain belong to the blue shifted progression of Cu2-Hen(L). Again, the liquid spectrum can be discerned in the baseline of the spectrum before the fountain, signifying molecules in droplets. In this case, the dimer spectra effectively terminate at the fountain, and the fountain can be directly visualized in the excitation image (Figure 9a).

This is re-enforced in the density profile of fluorescers, taken as a spatial cut across the fountain (Figure 9b). The fluorescence peak at the fountain clearly demonstrates build-up in the local den-

19 sity of Cu2,andthereforeentrainment.Nosuchfeaturecouldbeseeninthe nondivergent flow of Figure 7c. The spike in Figure 9b is over a smoothly decaying density profile, suggesting that a significant fraction is transmitted through the fountain. Subtraction of the background distribution with fountain off, shows a small deficit of liquid dimers around the fountain. There is no evidence of reflection of the molecules carried by the droplets at the interface, as there was in the case of the atoms.

Entrainment in the columnar fountain

Copper dimers can be completely entrained in the faster columnflow.Weillustratethisthrough spatial images recorded at selective wavelengths, designedtoenhancecontrastbetweentheliquid spectra and bare dimers. In Figure 10a we show a fluorescence image recorded with 438 nm exci- tation, in which the emission from entrained molecules from the fountain can be seen in the shape of the column. The excitation is shifted from the diatomic resonances, and the ablation intensity is adjusted to optimize loading of the fountain. Cuts at several delays and wavelengths are shown in Figure 10b and Figure 10c, respectively. These are obtained after subtracting the background signal obtained in the absence of the fountain. The dimer distribution between target and fountain can be seen in addition to the entrained molecules. In sharp contrast with the misty fountain, now the backside of the fountain is dark – trapped molecules cannot exit the sharp interface. The cross- sectional set recorded as a function of excitation delay, gives a qualitative picture of the velocity dependent trapping probabilities. In the case shown, the maximal loading occurs at a delay of 2 ms.

20 Figure 9: Fountain in misty mode operation. a) Two-dimensional excitation-emission spectrum (x-axis corresponds to the distance from the ablation target; y-axis is the excitation wavelength; contour colors denote the fluorescence intensity in arbitrary units with green and red representing high and low intensity, respectively), b) horizontal cut along the line shown in (a) represents the distance dependent distribution of the fluorescing dimers. The peak at the fountain, at 1.5 mm from target, indicates entrainment. c) Spectral cuts before and after the fountain (at 1 mm and 3 mm, respectively) show the complete blending of the diatomic lines into the broad (liquid) spectrum. The surviving sharp lines between 435 and 440 nm are from linear complexes.

21 Figure 10: a) Spatial fluorescence image demonstrating the loading of the columnar fountain (exci- tation at 438.2 nm; contour colors represent the emission intensity in arbitrary units with green and light blue corresponding to high and low intensity, respectively); b) Time evolution of the fountain loading (maximum loading at 2000 µs), given by the delay time between ablation with the excimer pulse and interrogation with the dye laser pulse; c) Cu2 fluorescence image inside the fountain at selected excitation wavelengths.

22 Discussion

Copper dimer

Although there has been extensive spectroscopy on Cu2,theexistingtheoreticalanalysisofthe excited states is limited.55,56 It is therefore useful to consider the spectroscopically relevant molec- ular potentials in a consistent calculation. The calculatedlow-lyingexcitedelectronicsingletΣ, Π and Δ states of Cu2 are shown in Figure 11. It was necessary to include the DKH Hamiltonian in the calculation with the appropriate DKH compatible basis set to obtain correct energetics for the states. In addition to serious convergence issues, the attempted multi-reference configuration in- teraction calculations were extremely sensitive to the active space chosen and the associated rapid increase in the computational demand. The EOM-CCSD method, on the other hand, was more robust in converging to the correct states around the equilibrium distance but due to the single reference nature of the method, it failed after ca. 3ÅdistanceastheT1 norm increased rapidly exceeding the suggested value in the literature (0.02).36 The radiative lifetimes of the B - X and

C-Xtransitionswereevaluatedfromthetransitiondipolemoments between states and Einstein coefficient of spontaneous emission. For the A state only a single point calculation was carried out since the computational expense of UHF based EOM-CCSD is fairly high and the current ex- periments focused on the X, B and C states. The effect of the spin-orbit coupling to mix the A and B states has a negligible effect on the energetics of the states (Eq. (1)) but allows the A state to borrow intensity from the B state through the mixing. This makes the A - X transition weakly electric dipole allowed. The relevant potential parametersfortheX,A,BandCstatesaresum- marized in Table 1 and a comparison between the calculated andavailableexperimentaldatais provided. The crossing point between the B and C states lies almost directly above the X ground state giving a high degree of overlap between B and C state vibronic wavefunctions. In the present measurements, the C state rapidly converts to B and the fluorescence is observed mainly from the Bstate.Thecalculatedequilibriumbondlengths,electronic origins and radiative lifetimes are in fair agreement with the experimental data, the harmonic vibrational frequencies are only of the

23 correct magnitude but the relative orders between the B and C states are reversed. The most likely factor limiting the accuracy of the current calculations is the applied triple zeta level basis set. At the present level of accuracy, the calculations fully support the experimental assignment of states and observations at the reported resolution, of otherwise well-known spectroscopy.29–33

4

3

1 3 Σg (X) Πu (A) part + 1 of 0 u with B Σg 1 Δg E (ev) E 2 μ = 0.46 a.u. μ = 1.66 a.u. 1 tr tr Πu (C) τ = 1 μs (C) τ = 200 ns (B) 1 Σu (B) 1 Δu 1 1 Πg

0 2 2.2 2.4 2.6 2.8 3 R (Angs)

Figure 11: Ground and several excited states of Cu2 from RHF-EOM-CCSD calculations. The A state was obtained from a single point calculation employing the UHF-EOM-CCSD method.

Copper atom scattering at the liquid interface

The theoretically extracted Cu-He potential is shown in Figure 12 and its parametrization is given

1 in Table 2. The interaction is weak, characterized by a potential well depth of 4.8 cm− ,slightly more than half the depth of the He-He potential.57 Energetics in such shallow potentials will be sensitive to the long-range part of the interaction, which inthepresentisoflimitedreliability. As such, the fitted form should be used with caution. The parametrized potential supports one bound rotationless state and one bound state in the J = 1 centrifugal potential (see Table 3). Cu

24 can be regarded as “heliophobic”, and as such is expected to prefer surface attachment rather than penetrate to the bulk.58 This, we verify explicitly, through bosonic DFT calculations of the energetics of Cu on a droplet with 1000 helium atoms. Due to the summation of the van der Waals interaction, the binding energy of Cu on the surface of the droplet is 22 K. Nevertheless, the atom is unstable when embedded in the droplet – the free energy of the system is minimized with the atom bound to the surface. This can be understood as the effectofminimizingthecavityenergyin afinitesizedroplet.

6 8 Table 2: Parametrized potentials fitted to the form: V(r)=a0 exp( a1r) a2r− a3r− ,in 1 − − − Å-cm− units (for fitting purposes only). rmin and Vmin correspond to the minimum energy distance and energy, respectively. T and L denote the linear and T-shaped geometries.

Cu-He L-Cu2(X)-He T-Cu2(X)-He L-Cu2(B)-He T-Cu2(B)-He a 1.1 106 5.0 106 2.1 106 3.6 106 2.2 106 0 × × × × × a1 2.544 2.446 2.590 1.973 2.814 a 7.661 104 7315 1.236 105 39.81 1.326 105 2 × × × a 1.079 106 1.383 107 1.723 106 3.357 107 8.878 105 3 × × × × × rmin 4.8 5.0 4.8 7.2 4.1 Vmin 4.6 11.7 7.9 217.8 !

100.0 T approach Cu2(X) 80.0 L approach Cu2(X) L approach Cu2(B) 60.0 T approach Cu2(B) Cu - He ground state

K) 40.0 E (

20.0

0.0

-20.0 5.0 10.0 R (Angs)

Figure 12: Calculated interaction potentials between Cu-He (CCSD(T)), Cu2(X)-He (CCSD(T)) and Cu2(B)-He (EOM-CCSD).

25 The most important observation regarding Cu atoms in the present is their scattering dynamics

at the fountain interface. We observe the scattering dynamics in 2Kisotropiccollisionsat ! the surface to be dominated by reflection. Given a surface binding energy that is an order of magnitude larger than the kinetic energy of collision, in classical gas-liquid scattering, efficient trapping and negligible reflection would be expected.59 In classical scattering, multiple collisions at the dynamically disordered surface would ensure energy loss, especially where the collision energy is the same as the temperature of the liquid. Although to date limited to helium atoms, averydifferentpicturecontrolsquantumscatteringatthegas-superfluid interface. The classic reflection measurements of helium atoms from the free surfaceofsuperfluidhelium60,61 confirmed the earlier suggestions by Anderson62 and Widom63,64 that quantum evaporation or scattering is controlled by a single particle event, the creation or annihilation of a single phonon or roton. This occurs on the attractive part of the interaction potential, in contrast to classical scattering, which is dominated by hard-core collisions. At any angle of incidence a Cu atom can create a 2 K phonon to fall into the bound part of the Cu-He(Liq) potential.However,thisdoesnotconstitute condensation. From such a precursor state, either additional excitations have to be generated to trap or the atom can re-evaporate with the annihilation of a thermally populated phonon (or roton). With detailed balance in mind, it is clearer to consider the evaporation of a bound Cu atom. The excitation must now overcome the binding energy of the atom, V = 22 K, and supply the additional kinetic energy, T,toreachtheinitialscatteringstate.SinceT +V exceeds that of the maxon (14.5 K),6 the dispersion curve of the heavy atom, E = p2/2m+V,doesnotcrossthatofhelium.Neither

quantum evaporation nor quantum condensation is possible, leaving elastic scattering as the main channel for the Cu atom interaction with the helium fountain.

Copper dimer - helium interaction and complexes

The interaction of ground state Cu2 molecule with helium was calculated at the CCSD(T)/AV5Z level to provide an optimal account for treating electron correlation, which is essential in modeling

van der Waals interactions. The results are shown in Figure 12forlinear(“L”)andbroadside(“T”)

26 approaches of helium. The linear geometry is more bound (17 K)thantheTstructure(11K).

For a freely rotating Cu2 molecule in helium, in J = 0state,itismorerelevanttoconsiderthe rotationally averaged potential, which has a well depth of 13K.Wecomputetheboundrotational

vibrational states of the L- and T-structures, subject to theadditionalcentrifugalpotential:

J(J + 1) V(r,J)=V(r)+ (2) 2µr2

The bound states and their energies are listed in Table 3. The results are subject to the same

cautionary comments made in the Cu-He discussion – the long-range part of the potential can significantly alter the obtained binding energies. This willnotchangethefindingsthatintheT-

complex the binding of He to Cu2 is stronger in the B state (see potentials in Figure 12); and that the vibrational level spacings in X and B states are quite similar (see Table 3). At T = 2K,several rotation-vibration states should be populated in the groundstateandasharplinespectrumisto

be expected in these bound-bound transitions. The perturbedhigh-resolutionlinespectrumseen at T = 2KinFigure5isonlyconsistentwiththeT-structuredcomplex. The spectrum contains one prominent sharp line accompanying each isotopic origin of bare Cu2 (the line not reproduced 1 by the simulation), with a blue shift of 0.16 cm− .Onlyoneprominentlineabovetheoriginis expected for the complex, namely the v = 0, J = 1 v = 0, J = 0transition,forwhicha " " ← "" "" 1 blue shift of 1.1 cm− is predicted (see Table 3). We take this qualitative agreement as basis for

assigning the spectrum to the T-shaped complex (T-Cu2-He). In the linear 1:1 complex, the excited B state potential does not sustain a bound state (Ta-

ble 3). The vertical energy difference, ΔV(r )=V(B) V(X),atthepotentialminimumofthe min − 1 ground state is 100 cm− . To rationalize the spectrum assigned to Cu2-Hen(L), which appears as 1 acompletesequenceshiftedby 600 cm− (Figure 6a), at least six such repulsive interactions are ! needed. Note that for broadside attachment the vertical energy difference is negligible, the present

1 potential would account for only 200 cm− blue shift for a shell of He atoms, derived mainly ! from the two axial atoms He-Cu2-He. This brings into question the accuracy of the potentialson

27 Table 3: Rotation-vibration states of Cu(1,2)-He. Cu-He 1 vJcm− 00-0.86 1-0.016

L-Cu2(X)-He T-Cu2(X)-He T-Cu2(B)-He 1 1 1 vJcm− vJcm− vJcm− 00-12.8 0 0 -7.77 0 0 -8.22 1-11.7 1 -6.63 1 -6.64 2-9.64 2 -4.39 2 -3.64 3-6.54 3 -1.16 4-2.52

10-2.11 1 0 -0.53 1 0 -0.28 1-1.47 1 -0.04 2-0.26

20-1.210 5 × − the repulsive wall; however, does not change the qualitativepredictions.Theblueshiftedprogres- sion assigned to Cu2-Hen(L) must be associated to linear complexes that fall apart in the excited state to release the linearly bound He atoms. It is useful to estimate the timescale of predissociation by considering the time it takes for the overlap of the ground and excited states wavefunctions to decay along the Cu2-He coordinate. This is defined by the time correlation function:

2 t/τpre X B 2 X iHBt/h X c(t)=e− = ϕ (t = 0) ϕ (t) = !$ϕ (t = 0) e− ϕ (t = 0)%! (3) !" | #! ! | ! ! ! ! !

in which the time propagation is on the excited state L-Cu2(B)-He potential. A predissociation time

of τpre = 2.8psisobtainedusingthepresentpotentials.Thistimescalealsodefinestheminimum 1 convolution width of 12 cm− that will contribute to any line transitions in the complex. This & explains the absence of sharp structure in Cu2-Hen(L) spectra inspected at high resolution. We

rationalize the blue shifted progression as arising from Cu2 linearly attached to at least one helium atom; and based on the energetics, it is more likely to have both axial wells occupied. This would

include the possibility for a complete shell of helium atoms enclathrating the dimer.

28 The spectra obtained as a function of Cu2 transport through the dense gas show a very different behavior. The dry clusters show blue tails that evolve continuously as the cluster grows, but without ashiftofvibrationalorigins(seesharpedgesinFigure8a).SincealinearlyboundHeatomwill

1 lead to > 100 cm− blue shift, it becomes apparent that the dry clusters correspond to structures in which the copper dimer lies flat on the surface of a growing Hecluster.Above,wemadethis conclusion regarding Cu atoms. The broadening of the vibrational origins (Figure 8b), is then consistent with the dimer being solvated in the cluster, “wetting”, whereby it is subject to both linear and T-interactions, as in our discussion of a full solvation shell shifting the electronic origin

1 by 600 cm− .ThebindingenergyofthedimertoHeislargerthanfortheatom and since it exceeds the cohesive energy of helium, we would expect dimerstoembedinthelargerclusters.

This is supported by the bosonic DFT calculations, which predict that Cu2 should reside inside helium droplets (the free energy of solvation nearly 150 K foradropletwith500Heatoms).The experiment may be selective in identifying the surface boundmolecules,whichwouldbethecase if, for example, non-radiative decay dominates the excited state relaxation in the bulk. Clearly, the excitation spectra of the dimer are strongly perturbed by attachment, or solvation in liquid helium.

This is in stark contrast with the published LIF spectra of Cu2,obtainedbylaserablationinthe bulk liquid and solid as a function of pressure where unperturbed molecular vibronic transitions are observed in emission.16,19 Presumably, the emission there occurs in large enough bubbles to be completely impervious to the local helium environment. Given the large free energy of solvation of Cu2 in helium, the observation that in general the transport of the cold molecules through the fountain is unimpeded (Figure 7 and Figure 9) is nontrivial, even though formally, there is no mechanism for scattering of particles that travel at speeds below the critical velocity of helium.

Conclusions

We described the realization of molecular injection in a superfluid helium fountain. The results are intriguing and promise a variety of applications with this uniquely flexible source and method.

29 The utility for spectroscopic applications was highlightedthroughtheanalysisofthedimerspec- tra. A more rigorous treatment of the spectra will require numerical modeling. The platform also provides direct investigations of scattering processes, along which we gave two outstanding exam- ples: quantum reflection of the atom, and unimpeded transmission of dimers through the fountain. These results are experimental observations that will require further scrutiny to be fully appreci- ated. Finally, we clearly show entrainment of molecules in a directed, non-divergent flow as a beam. Beyond the fundamental questions, entraining molecules in a homogeneous cold column opens up diverse applications. Focusing of the molecular column, for lithography or for homoge- neous condensation,65 bending and manipulating the beam by irradiation with an inhomogeneous intensity field of a laser, are examples.

Acknowledgement

Financial support from the National Science Foundation: grants CHE-0949057 to JE and CHE- 0802913 to VAA, and Graduate School of Computational ChemistryandSpectroscopy(LASKEMO;

Finland) grant to EV are gratefully acknowledged. Additional computational resources were pro- vided by National Center for Supercomputing Applications (NCSA) TeraGrid grant TG-CHE100150.

References

(1) Toennies, J. P.; Vilesov, A. F. Rev. Phys. Chem. 1998, 49,1.

(2) Toennies, J. P.; Vilesov, A. F. Angewandte Chemie (Intl. Ed.) 2004, 43,2622.

(3) Toennies, J. P. Physik Journal 2002, 1,49.

(4) Goyal, S.; Schutt, D. L.; Scoles, G. Phys. Rev. Lett. 1992, 69,933.

(5) Khalatnikov, I. M. The Physics of Liquid and Solid Helium;Wiley:NewYork,1976.

(6) Wilks, J. The properties of liquid and solid helium; Clarendon Press: Oxford, 1967.

30 (7) Hartmann, M.; Miller, R. E.; Toennies, J. P.; Vilesov, A. F. Phys. Rev. Lett. 1995, 75,1566.

(8) Ghazarian, V.; Apkarian, J. E. V. A. Rev. Sci. Instr. 2002, 73,3606.

(9) Slipchenko, M. N.; Kuma, S.; Momose, T.; Vilesov, A. F. Rev. Sci. Instr. 2004, 73,3600.

(10) Claas, P.; Mende, S.-O.; Stienkemeier, F. Rev. Sci. Instr. 2003, 74,4071.

(11) Pentlehner, D.; Riechers, R.; Dick, B.; Slenczka, A.; Even, U.; Lavie, N.; Brown, R.; Luria, K. Rev. Sci. Instr. 2009, 80,043302.

(12) Bierau, F.; Kupser, P.; Meijer, G.; von Helden, G. Phys. Rev. Lett. 2010, 105,133402.

(13) Jortner, J.; Meyer, L.; Rice, S.; Wilson, E. G. Phys. Rev. Lett. 1964, 12,415.

(14) Dennis, W. S.; E. Durbin, J.; Fitzsimmons, W. A.; Heybey,O.;Walters,G.K.Phys. Rev. Lett. 1969, 23,1083.

(15) Kinoshita, T.; Fukuda, K.; Yabuzaki, T. Phys. Rev. B 1996, 54,6600.

(16) Persson, J. L.; Hui, Q.; Nakamura, M.; Takami, M. Phys. Rev. A 1995, 52,2011.

(17) Hui, Q.; Takami, M. J. Low Temp. Phys. 2000, 119,393.

(18) Tabbert, B.; Beau, M.; Gunther, H.; Haussler, W.; Honninger, C.; Meyer, K.; Plagemann, B.; zu Putlitz, G. Z. Phys. B 1995, 97,425.

(19) Lebedev, V.; Moroshkin, P.; Toennies, J. P.; Weis, A. J. Chem. Phys. 2010, 133,154508.

(20) Allen, J. F.; Jones, H. Nature 1938, 141,243.

(21) Brown, M.; Wyatt, A. F. G. J. Phys: Condens. Matter 1990, 2,5025.

(22) Williams, C. D. H. J. Low Temp. Phys. 1998, 113,11.

(23) Dalfovo, F.; Guilleumas, M.; Lastri, A.; Pitaevski, L.;Stringari,S.J. Phys. Condens. Matter 1997, 9,L369.

31 (24) Vertes, A.; Dreyfus, R. W.; Platt, D. E. IBM Journal of Research and Development 1994, 38,

3.

(25) Autin, M.; Briand, A.; Mauchien, P.; Mermet, J. M. Spectrochimica Acta 1993, 48B,851.

(26) Dreyfus, R. W. J. Appl. Phys. 1991, 69,3.

(27) Gole, J. L.; English, J. H.; Bondybey, V. E. J. Phys. Chem. 1982, 86,2560.

(28) Bondybey, V. E.; Schwartz, G. P.; English, J. H. J. Chem. Phys. 1983, 78,11.

(29) Page, R. H.; Gudeman, C. S. J. Chem. Phys. 1991, 94,39.

(30) Lochet, J. J. Phys. B 1978, 11,L55.

(31) Rohlfing, E. A.; Valentini, J. J. J. Chem. Phys. 1986, 84,6560.

(32) Ram, R. S.; Jarman, C. N.; Bernath, P. F. J. Mol. Spectr. 1992, 156,468.

(33) McCaffrey, J. G.; Bennet, R. R.; Morse, M. D.; Breckenridge, W.H.J. Chem. Phys. 1989,

91,92.

(34) Eloranta, J. A Simplified C Programming Interface for Research Instruments, http://sourceforge.net/projects/libmeas/. 2010.

(35) Korona, T.; Werner, H.-J. J. Chem. Phys. 2003, 118,3006.

(36) Lee, T. J.; Taylor, P. R. J. Quantum Chem. 1989, S23,199.

(37) Reiher, M.; Wolf, A. J. Chem. Phys. 2004, 121,2037.

(38) Reiher, M.; Wolf, A. J. Chem. Phys. 2004, 121,10945.

(39) Wolf, A.; Reiher, M.; Hess, B. A. J. Chem. Phys. 2002, 117,9215.

(40) Balabanov, N. B.; Peterson, K. A. J. Chem. Phys. 3005, 123,064107.

32 (41) Werner, H.-J. et al. MOLPRO, version 2010.1, a package ofabinitioprograms.2010;see

http://www.molpro.net.

(42) Ermler, W. C.; Lee, Y. S.; Pitzer, K. S. J. Chem. Phys. 1979, 70,293.

(43) Frisch, M. J. et al. Gaussian 09 Revision A.1. Gaussian Inc. Wallingford CT 2009.

(44) Hampel, C.; Peterson, K.; Werner, H.-J. Chem. Phys. Lett. 1992, 190,1.

(45) Deegan, M. J. O.; Knowles, P. J. Chem. Phys. Lett. 1994, 227,321.

(46) Woon, D. E.; T. H. Dunning, J. J. Chem. Phys. 1994, 100,2975.

(47) Boys, F.; Bernardi, F. Mol. Phys. 1970, 19,553.

(48) Dalfovo, F.; Lastri, A.; Pricaupenko, L.; Stringari, S.; Treiner, J. Phys. Rev. B 2005, 72,

214522.

(49) Lehtovaara, L.; Toivanen, J.; Eloranta, J. J. Comp. Phys. 2007, 221,148.

(50) Lehtovaara, L.; Kiljunen, T.; Eloranta, J. J. Comp. Phys. 2004, 194,78.

(51) Fuhr, J. R.; Wiese, W. L. CRC Handbook of Chemistry and Physics (86th Ed.); CRC Press: Boca Raton, FL, 2005.

(52) Ghazarian, V. Ph.D. Thesis; University of California: Irvine, 2005.

(53) Kelly, R.; Dreyfus, R. W. Surf. Sci. 1988, 198,263.

(54) Morse, M. D. Chem. Rev. 1986, 86,1049.

(55) Witko, M.; Beckmann, H.-O. Mol. Phys. 1982, 47,945.

(56) Danovich, D.; Filatov, M. J. Phys. Chem. A 2008, 112,12995.

(57) Janzen, A. R.; Aziz, R. A. J. Chem. Phys. 1997, 107,914.

33 (58) Ancilotto, F.; Lerner, P. B.; Cole, M. W. J. Low Temp. Phys. 1995, 101,1123.

(59) Nathanson, G. M. Rev. Phys. Chem. 2004, 55,231.

(60) Edwards, D. O.; Fatouros, P.; Ihas, G. G.; Mrozinski, P.;Shen,S.Y.;Gasparini,F.M.;

Tam, C. P. Phys. Rev. Lett. 1975, 34,1153.

(61) Edwards, D. O.; Fatouros, P. P. Phys. Rev. B 1977, 17,2147.

(62) Anderson, P. W. Phys. Lett. A 1969, 29,563.

(63) Widom, A. Phys. Lett. A 1969, 29,96.

(64) Hyman, D. S.; Scully, M. O.; Widom, A. Phys. Rev. 1969, 186,231.

(65) Gordon, R. J.; Zhu, L.; Schroeder, A.; Seideman, T. J. J. Appl. Phys. 2003, 94,669.

34