Electronic Structure of Rare Gas - Hydrogen Rydberg Molecules Erich - - PowerPoint PPT Presentation

Introduction Results Acknowledgement

Electronic Structure of Rare Gas - Hydrogen Rydberg Molecules

Erich W Schreiner and Geerd HF Diercksen

Max-Planck-Institut für Astrophysik Garching, GERMANY

November 13, 2007

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Hydrogen spectrum

The Gymnasiallehrer J. J. Balmer from Basel, Switzerland, was the first to recognize that the spectrum of the hydrogen atom consists of series of spectral lines and that the frequencies of the lines can be calculated by the formula ν = ν0 − R n2 R: Rydberg constant.

Reference: J. J. Balmer, Notiz über die Spektrallinien des Wasserstoffs, Verhandlungen der Naturforschenden Gesellschaft in Basel, 7, 552 (1885)

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Hydrogen spectrum

✻ E

l = 0 l = 1 l = 2 l = 3 n=1 n=2 n=3

✟ ❍ ❏ ❏▲ ▲ ▲

n=4 n=5 n=6 n=7 Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Balmer

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Balmer

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Many-electron atom

The Swedish physicist J. R. Rydberg (1854–1919) was the first to recognize that “the spectra of [many-electron] atoms can be understood much in the same way as the spectrum of the Hydrogen atom” and that the frequencies of the spectral lines can be computed by the formula: ν = ν0 − R (n − δ)2 R: Rydberg constant.

Reference: J. R. Rydberg, On the structure of the line-spectra of the chemical elements, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 29, 331 (1890)

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Many-electron atom

The expectation value for the radius of a Rydberg state is given by the formula r = 3n2 − l(l + 1) 2Z a0

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Many-electron spectrum

✻ E

l = 0 n=1 n=2 n=3

✟ ❍ ❏ ❏▲ ▲ ▲

n=4 n=5 n=6 n=7 ⊕ proton n=1 1.5 Bohr n=2 6.0 Bohr n=3 13.5 Bohr n=4 24.0 Bohr n=5 37.5 Bohr n=100 15000 Bohr n=200 60000 Bohr

✻ r

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Rydberg

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Rydberg

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Rydberg

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Rydberg

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Characteristics

• Chemically unbound ground state

Chemically bound excited states Large energy gap between ground and excited states

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Outline

Schrödinger equation Configuration interaction (CI) method Confining potential Gaussian basis set

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Schrödinger equation

[H(r)] Ψ(1, 2, . . . , N) = EΨ(1, 2, . . . , N) H(r) =

N

• i=1
• − 1

2∇2 i

• +

N

• i=1

M

• α=1
• −

Zα |ri − Rα|

• +

N

• i=1

w(ri) +

N

• i>j
• 1

|ri − rj|

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Wavefunction approximation

|Ψ = |Φ = Φ(x1x2 · · · xN) = (N!)− 1

2

χi(x1) χj(x2) · · · χk(xN)

• χ =

ψ · α ψ · β ψi =

• m

cimξm |Ψ = C0|Φ0 +

• ra

Cr

a|Φr a +

• a<b

r<s

Crs

ab|Φrs ab +

• a<b<c

r<s<t

Crst

abc|Φrst abc + · · ·

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Basis set conditions

The basis functions have to describe/model at the Rydberg center:

the electron distribution of the positive ion the electron distribution of Rydberg states the lectron distribution in the external potential, if present

at the ligands:

the electron distribution of the neutral system the electron polarisability the lectron distribution in the external potential, if present

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

HeH+: Spectroscopic constants

Table: Spectroscopic constants of HeH+ in the 1Σ+ state without confinement (re in atomic units, νe and νexe in cm−1) Reference re νe νexe He-21/H-36 1.4770 3177 141 He-21/H-36 1.4694 3275 149 He-21/H-55 1.4765 3201 145 He-34/H-55 1.4732 3206 146 Harris 1.444 3379 314 Wolniewicz 1.4632 3233 617 Kolos 1.4632 3220 166 Calculated by fitting Wolniewicz and Kolos’ ab initio potential energy curves to Dunham’s 4-th order polynomial.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

HeH: Spectroscopic constants

Table: Spectroscopic constants of HeH without confinement

State Reference re νe νexe De A 2Σ+ He-21/H-18 1.4077 3670 255 2.53 He-21/H-36 1.4118 3699 196 2.54 He-21/H-55 1.4149 3722 183 2.54 He-34/H-55 1.4120 3726 182 2.54 Sarpal 1.4040 3512 Buenker 1.4115 3662 2.48 Ketterle 1.43 3701 Ketterle 1.4003 3718 161 B 2Π He-21/H-18 1.4634 3364 199 2.21 He-21/H-36 1.4629 3330 205 2.18 He-21/H-55 1.4652 3367 201 2.19 He-34/H-55 1.4622 3372 199 2.20 Sarpal 1.4571 3158 Buenker 1.4629 3302 2.20 C 2Σ+ He-21/H-18 1.5413 2928 210 1.64 He-21/H-36 1.5409 2930 190 1.61 He-21/H-55 1.5417 3007 199 1.61 He-34/H-55 1.5370 2957 211 1.61 Sarpal 1.5255 2788 Buenker 1.5428 2872 1.65 Ketterle 1.57 2896 Ketterle 1.5324 2902 141 Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

HeH: Spectroscopic constants

Table: Spectroscopic constants of HeH without confinement (re in atomic units, νe and νexe in cm−1, De in eV)

State Reference re νe νexe De D 2Σ+ He-21/H-18 1.4545 3418 196 2.18 He-21/H-36 1.4479 3467 289 2.16 He-21/H-55 1.4584 3402 202 2.15 He-34/H-55 1.4555 3405 201 2.16 Sarpal 1.4504 3187 Buenker 1.4508 3383 2.14 E 2Π He-21/H-18 1.4719 3296 204 2.09 He-21/H-36 1.4725 3258 193 2.08 He-21/H-55 1.4736 3297 207 2.07 He-34/H-55 1.4708 3299 207 2.08 Sarpal 1.4655 3083 Buenker 1.4718 3233 2.09 F 2Σ+ He-21/H-18 1.5518 3013 209 1.46 He-21/H-36 1.4696 3276 303 2.03 He-21/H-55 1.4801 3246 209 2.05 He-34/H-55 1.4771 3252 208 2.06 Sarpal 1.4693 3057 Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

NeH+: Spectroscopic constants

Table: Spectroscopic constants of NeH+ without confinement. (re in atomic units, De in eV, and νe and νexe in cm−1) State Reference re νe νexe De NeH+ X 1Σ+ This work 1.8628 2953 135.8 2.37 Bondybey 1.8689 2917 2.10 Hayes 1.876 2910 105.9 2.33 Hayes 1.882 2894 116.9 2.27 Hayes 1.876 2892 114.5 2.29 Expt 1.8727 2904 113.4 2.28

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

NeH: Spectroscopic constants

Table: Spectroscopic constants of NeH without confinement. (re in atomic units, De in eV, and νe and νexe in cm−1)

State Reference re νe νexe De A 2Σ+ This work 1.8894 2869 136.7 2.03 Baer 1.9067 2855 108.4 2.04 Bondybey 1.8822 2801 1.53 Buenker 1.9332 2727 2.01 B 2Π This work 1.8681 2952 167.8 1.86 Baer 1.8671 2883 124.0 1.84 Bondybey 1.8671 2913 1.50 Buenker 1.9162 2792 1.79 C 2Σ+ This work 1.9392 2780 129.0 1.50 Baer 1.9464 2646 95.6 1.37 Bondybey 2.0579 2737 1.51 Buenker 1.9842 2596 1.79 Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Re HeH = 1.3a0 Re NeH = 2.0a0 Re ArH = 2.6a0 Re KrH = 2.8a0

Figure: Total energy of the cations RgH+, Rg = He, Ne, Ar, Kr as function of the internuclear distance RRgH

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Re NeH = 2.0a0 Re ArH = 2.6a0 Re KrH = 2.8a0

Figure: Total energy of the cations RgH+, Rg = Ne, Ar, Kr as function of the distance RRgH with and without using pseudo-potentials

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Re HeH = 1.3a0 Re NeH = 2.0a0 Re ArH = 2.6a0 Re KrH = 2.8a0 Re XeH = 3.0a0

Figure: Total energy of the cations RgH+, Rg = He, Ne, Ar, Kr, Xe as function of the internuclear distance RRgH using pseudo-potentials

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Re HeLi = 3.6a0 Re NeLi = 3.8a0 Re ArLi = 4.35a0 Re KrLi = 4.3a0

Figure: Total energy of the cations RgLi+, Rg = He, Ne, Ar, Kr as function of the internuclear distance RRgLi

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Binding energies and bond length

Table: Calculated binding energies (in mHartree = 0.68 kcal/mol = 27 meV) and bond lengths (in a0 = 0.52 Å)

He Ne Ar Kr H HeH+ 74 NeH+ 91 ArH+ 150 KrH+ 207 Li HeLi+ 2.8 NeLi+ 8.6 ArLi+ 9.8 KrLi+ 38 RRg He (2.3) Ne (3.0) Ar (3.6) Kr (3.7) H HeH+ 1.48 NeH+ 1.84 ArH+ 2.42 KrH+ 2.60 Li HeLi+ 3.60 NeLi+ 3.78 ArLi+ 4.47 KrLi+ 4.27 RH ≈ 0a0, RLi = 1.1a0 Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Excitation energies

Table: Excitation energies for the hydrogen and rare gas atoms (in Hartrees = 27 eV)

0.0 0.5 H He Ne Ar Kr Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Excitation energies

Table: Excitation energies for the lithium and rare gas atoms (in Hartrees = 27 eV)

0.0 0.5 Li He Ne Ar Kr Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

ground state n = 2 n = 3 n = 4 n = 5 cation

Figure: Total energy of various electronic states of HeH as function of the internuclear distance RHeH

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

n = 2 n = 3 n = 4 n = 5 cation

Figure: Total energy of various electronic states of HeH as function of the internuclear distance RHeH

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Total energy of various electronic states of ArH as function of the internuclear distance RArH

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Total energy of various electronic states of ArH as function of the internuclear distance RArH

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Total energy of various electronic states of KrH as function of the internuclear distance RKrH

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Total energy of various electronic states of KrH with the internuclear distance RKrH

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Total energy of various electronic states of XeH as function of the internuclear distance RXeH using pseudo-potentials

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Comparison of total the energy of various electronic states of HeH and KrH as function of the internuclear distance

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Comparison of the total energy of various electronic states of HeLi and ArLi as function of the internuclear distance

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

sσ states pσ states pπ states cation

Figure: Total energies of various electronic states of He-H-He at linear symmetric geometries as function of RHe1H = RHe2H

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Potential energy curves

Figure: Total energies of various excited states of He-H-He at bent symmetric geometries with RHe1H = RHe2H fixed at 2a0 and 4a0 as function of the bond angle

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Electron densities

• Erich W Schreiner and Geerd HF Diercksen

Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Confining potential

The confining potential is of the form of the cylindrical

• scillator potential:

Wω(ri) = ω2 2 (x2

i + y2 i ).

(1) The molecule is aligned along the z-axis

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Potential energy curves

0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 8 E - Elim (in au) R (in au)

Figure: Potential energy curves

• f selected low-lying electronic

states of HeH and of the ground state of HeH+. The solid and dotted lines are the potential energy curves of the 2Σ+ states

• f HeH and the dashed line of

the X 1Σ+ state of HeH+.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Potential energy curves

• 3.10
• 3.05
• 3.00
• 2.95
• 2.90
• 2.85
• 2.80

1 2 3 4 5 6 7 8 E (in au) R (in au) (a)

Figure: Potential energy curves

• f the excited A 2Σ+ state of

HeH in confinement. The bottom line corresponds to ω = 0.0 au and the successive lines represent increments in ω

• f 0.025 au.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Potential energy curves

1 2 3 4 5 6 7 8 R (in au) (b)

Figure: Potential energy curves

• f the excited B 2Π state of HeH

in confinement. The bottom line corresponds to ω = 0.0 au and the successive lines represent increments in ω of 0.025 au in ω.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Potential energy curves

• 3.10
• 3.05
• 3.00
• 2.95
• 2.90
• 2.85
• 2.80

1 2 3 4 5 6 7 8 E (in Eh) R (in au) (c)

Figure: Potential energy curves

• f the excited C 2Σ+ state of

HeH in confinement. The bottom line corresponds to ω = 0.0 au and the successive lines represent increments in ω

• f 0.025 au.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Potential energy curves

1 2 3 4 5 6 7 8 R (in au) (d)

Figure: Potential energy curves

• f the the ground state X 1Σ+

state of HeH+ in confinement. The bottom line corresponds to ω = 0.0 au and the successive lines represent increments in ω

• f 0.025 au.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Equilibrium internuclear distance

Table: Equilibrium internuclear distance re (in atomic units). (au) A 2Σ+ B 2Π C 2Σ+ D 2Σ+ E 2Π F 2Σ+ 0.000 1.4118 1.4629 1.5409 1.4573 1.4725 1.4800 0.025 1.4055 1.4622 1.5571 1.4686 1.4788 1.4546 0.050 1.3922 1.4594 1.5824 1.4661 1.4803 1.4478 0.075 1.3783 1.4558 1.6165 0.100 1.3660 1.4491 1.6521 0.125 1.3542 1.4435 0.150 1.3430

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Equilibrium internuclear distances

1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Equilibrium internuclear distance re (in au) Confining parameter ω (in au)

Figure: Variation of the equilibrium internuclear distances with the strength of the confinement. The distances are shown for the states A 2Σ+(plus), B 2Π+(fullsquare), C 2Σ+(times), D 2Σ+(opentriangle), E 2Π+(opencircle), and F 2Σ+(opensquare).

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Electron density difference

1 1 2 (a) H He

Figure: Electron density difference for the A 2Σ+ state. The solid contours indicate regions of increased electron density and the dashed contours regions of reduced electron density induced by the confinement with the strength ω = 0.025 au.

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HeH: Electron density difference

1 1 1 2 (b) H He

Figure: Electron density difference for the C 2Σ+ state. The solid contours indicate regions of increased electron density and the dashed contours regions of reduced electron density induced by the confinement with the strength ω = 0.025 au.

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HeH: Electron density difference

1 1 2 2 (c) H He

Figure: Electron density difference for the D 2Σ+ state. The solid contours indicate regions of increased electron density and the dashed contours regions of reduced electron density induced by the confinement with the strength ω = 0.025 au.

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Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Electron density difference

1 1 2 (d) H He

Figure: Electron density difference for the F 2Σ+ state. The solid contours indicate regions of increased electron density and the dashed contours the regions of reduced electron density induced by the confinement with the strength ω = 0.025 au.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Electron correlation energy

0.001 0.002 0.003 0.004 0.005 0.006 1 1.5 2 2.5 3 Ecorr (ω > 0) - Ecorr (ω = 0) (in au) R (in au)

Figure: Change in the electron correlation energy for the B 2Π

• state. The bottom line

corresponds to Ecorr(ω = 0.025) − Ecorr(ω = 0.0) and the lines above it represent increments in ω of 0.025 au.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Transition dipole moments

0.5 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 4.5 5 Transition dipole moment (in au) R (in au) ω = 0.00 ω = 0.00 ω = 0.05 ω = 0.05 ω = 0.10 ω = 0.10

Figure: Transition dipole moments as functions of internuclear distance R. Solid lines correspond to A-B transitions and dashed lines correspond to B-C transitions.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Oscillator strengths

0.00 0.05 0.10 0.15 0.20 Oscillator Strength f (a) ω = 0.00 ω = 0.05 ω = 0.10 ω = 0.00 ω = 0.05 ω = 0.10

Figure: Oscillator strengths for transitions between excited states as functions of the internuclear distance R. Solid lines correspond to X-A transitions and dashed lines correspond to X-C transitions.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Oscillator strengths

0.00 0.05 0.10 0.15 0.20 1 1.5 2 2.5 3 3.5 4 4.5 5 Oscillator Strength f R (in a.u.) (b) ω = 0.00 ω = 0.05 ω = 0.10 ω = 0.00 ω = 0.05 ω = 0.10

Figure: Oscillator strengths for transitions from the ground state as functions of the internuclear distance R. Solid lines correspond to X-A transitions and dashed lines correspond to X-C transitions.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: Critical confinement ωc

Table: Critical confinement parameter ωc for different electronic states

• f HeH

State ωc (au) A 2Σ+ 0.178 B 2Π 0.155 C 2Σ+ 0.111 D 2Σ+ 0.067 E 2Π 0.055 F 2Σ+ 0.042

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

HeH: ∆En(ω)

• 0.05

0.00 0.05 0.10 0.15 0.02 0.04 0.06 0.08 0.1 0.12 0.14 ∆ En (ω) (in au) Confinement parameter ω (in au)

Figure: ∆En(ω) for the six lowest electronic states of HeH as function of the confinement strength ωc. The lines correspond to the following values of n (states): n = 1 (A 2Σ+) (full); n = 2 (B 2Π+) (chain); n = 3 (C 2Σ+) (longbroken); n = 4 (D 2Σ+) (dashed); n = 5 (E 2Π+) (chain); n = 6 (F 2Σ+) (dotted)

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

NeH: Potential energy curves

• 129.3
• 129.2
• 129.1
• 129
• 128.9
• 128.8

1 2 3 4 5 6 7 8 E (in Eh) R (in a.u.) X 2Σ+ A 2Σ+ B 2Π C 2Σ+ Ne + H(1s) Ne + H(2s2p) Ne + H(3s3p3d) NeH+ X 1Σ+

Figure: Potential energy curves

• f selected low-lying states of

NeH and of the ground state of NeH+. (ω = 0.00 a.u.)

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

NeH: Equilibrium bond distances and vibrational frequencies

1.8 1.85 1.9 1.95 2 2.05 2.1 0.02 0.04 0.06 0.08 0.1 0.12 R (in a.u.) ω (in a.u.) A 2Σ+ B 2Π C 2Σ+

2200 2400 2600 2800 3000 3200 3400

νe (in cm-1) A 2Σ+ B 2Π C 2Σ+

Figure: Equilibrium bond distances and vibrational frequencies as function of the strengths of confinement for the first three low-lying electronic states of NeH.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

NeH: Potential energy curves

0.02 0.04 0.06 0.08 0.1 0.12 1 2 3 4 5 6 7 8 E (in Eh) ω = 0.00 NeH 2Σ+ NeH 2Π 0.02 0.04 0.06 0.08 0.1 0.12 1 2 3 4 5 6 7 8 ω = 0.04 NeH 2Σ+ NeH 2Π 0.02 0.04 0.06 0.08 0.1 0.12 1 2 3 4 5 6 7 8 E (in Eh) R (in a.u.) ω = 0.08 NeH 2Σ+ NeH 2Π 0.02 0.04 0.06 0.08 0.1 0.12 1 2 3 4 5 6 7 8 R (in a.u.) ω = 0.12 NeH 2Σ+ NeH 2Π

Figure: Potential energy curves

• f the 2Σ+ and 2Π states of NeH

in confinement. The energy is rescaled by Emin of the A 2Σ+ state as reference.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

NeH: Emin(NeH+) - Emin(NeH)

• 0.5

0.5 1 1.5 2 2.5 3 3.5 0.02 0.04 0.06 0.08 0.1 0.12 Emin(NeH+) - Emin(NeH) (in eV) ω (in a.u.) A 2Σ+ B 2Π C 2Σ+

Figure: Energy difference Emin(NeH+) - Emin(NeH) as function of the confinement strength ω. The energies are zero-point-energy corrected.

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

NeH: Critical confinement strength value ωc

Table: Critical confinement strength value ωc leading to

• autoionization. (ωc in atomic units)

State ωc A 2Σ+ 0.144 B 2Π 0.106 C 2Σ+ 0.107

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Acknowledgement

Outline

1

Introduction Atomic spectra Rydberg molecules Computational model outline Spectroscopic constants

2

Results Rare-gas hydrogen anions Rare-gas hydrogen molecules di-Helium hydrogen molecules Confined HeH structure and spectra Confined NeH structure and spectra

3

Acknowledgement Acknowledgement

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules

Introduction Results Acknowledgement Acknowledgement

Acknowledgement: Institutions

Erich W Schreiner and Geerd HF Diercksen Rydberg Molecules