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İSTANBUL TECHNICAL UNIVERSITY  IN STITUTE OF SCIENCE AND TECHNOLOGY

ACCURATE DETERMIN ATION OF THE POTENTIAL
ENERGY SURFACES AND THE MOLECULAR PROPERTIES
FOR MHe+ (M=Li, Na, K) CO MPLEXES USING QUANTUM
MECHANICA L TECHNIQUES

MS.SCI. Thesis by
Erol YILDIRIM, Chemist
Department : Chemistry
Program: Chemistry

Supervisor : Assoc. Prof. Dr. Mine YURTSEVER

MAY 2005
İSTANBUL TEKNİK ÜNİVERSİTESİ  FEN BİLİMLERİ ENSTİTÜSÜ

MHe+ (M=Li, Na, K) SİSTEMLERİ İÇİN POTANSİYEL ENERJİ
YÜZEYLERİNİN VE MOLEKÜL ÖZELLİKLERİNİN KUANTUM

MEKANİK YÖNTEMLERLE HASSAS OLARAK HESAPLANMASI

YÜKSEK LİSANS TEZİ
Erol YILDIRIM

Anabilim Dalı: Kimya
Programı: Kimya

Tez Danışmanı: Prof. Dr Mine YURTSEVER

HAZİRAN 2005

15
İSTANBUL TECHNICAL UNIVERSITY  I NSTITUTE OF SCIENCE AND TECHNOLOGY

ACCURATE DETERMINATIO N OF THE POTENTIAL
ENERGY SURFACES AND THE MOLECULAR PROPERTIES
FOR MHe+ (M=Li, Na, K) COMP LEXES USING QUANTUM
MECHANICAL TECHNIQUES

M.Sc Thesis by
Erol YILDIRIM, Chemist
(509031227)

Date of submission : 9 October 1996
Date of defence examination: 25 January 2005

Supervisor : Prof. Dr. Mine YURTSEVER
Members of the Examining Committee Ass. Prof. Fethiye Aylin KONUKLAR
Ass Prof. Nurcan TÜZÜN

JUNE 2005
ACKNOWLEDGEMENTS

I am grateful to my thesis supervisor, Prof. Dr. Mine YURTSEVER for her guidance,
helpful comments and critisizm throughout this research.

The computer time provided by High Performance Computing Center at ITU is
gratefully acknowledged.

JUNE 2005 Erol YILDIRIM

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TABLE OF CONTENTS
Page
LIST OF ABBREVIATIONS iv
LIST OF TABLES v
LIST OF FIGURES vi
LIST OF SYMBOLS vii
SUMMARY viii
ÖZET ix

1. INTRODUCTION 1
2. THEORY 8
2.1. Alkali ion-rare gas complexes 8
2.2. Elements and molecules in the He clusters 8
2.3. Special Properties of the ultra cold He 9
2.4. Structures formed by charged particles in the He 10
2.5. Doping Liquid He droplets 12
2.5.1. Droplet Sources 12
2.5.2. Doping Process 14
2.5.3. An example for the apparatus 16
2.6. Theoretical Studies 18
3. METHODS 21
3.1. Programmes and Technical supports used in calculations 21
3.2. Methods and basis sets used in calculations 23
3.2.1 Post-HF-Methods-Consideration of the Electron Correlation 23
3.2.1.1. Møller-Plesset Perturbation Theory 24
3.2.1.2. Configuration Interaction 28
3.2.1.3. Coupled Cluster Theory 32
3.2.2. Comparison of ab initio methods and Limitations 35
3.2.3. A Hybrid DFT method 38
3.2.4. Basis Sets 38
4. RESULTS 40
4.1. Moller Plessed Calculations 40
4.2. QCISD(T) and CCSD(T) 50
4.3. DFT/ B3LYP Calculations 53
4.4. Comparison of Results 54
4.4. Bound vibrational levels and charges 60
5. CONCLUSION 63
REFERENCES 65
BIOGRAPHY 75

iii
LIST OF ABBREVIATIONS

LHe : Liquid Helium
SWF : Shadow Wave Functions
M : Metal
Rg : Rare Gas
PEC : Potential energy Curve
MCS : Monte Carlo Simulation
MP : Moller Plessed
CC : Coupled Cluster
CI : Configuration Interaction
CCSD : Coupled Cluster including single and double substitutions
QCISD : Quadratic Configuration Interaction including singlets, doublets
G03 : Gaussian 2003
DFT : Density Functional Theory
HF : Hartree Foch
UHF : Unrestricted Hartree Foch
B3LYP : Becke Three Parameter Hybrid Functionals
cc-pvXz :Correlation Consistent Polarized Valence X (X=double, triple…)
Zeta basis set
CEPA : Coupled electron-pair approximation
IEPA : Independent electron-pair approximation
SCF : Self Consistent Field
SDTQ : Single, Double, Triple and Quadruple
BSSE : Basis Set Superposition Error
CPU : Central Processing Unit
H : Hamiltonian
Ψ : Wave function
ε : Eigenvalue
SAPT : Symmetry-adapted perturbation theory
PEC : Potential Energy Curve

iv
LIST OF TABLES

Page No
Table 4.1. D and R calculated for MHe+ by using MP2 method. 44
e e
Table 4.2. D and R calculated for MHe+ by using MP2=full method. 47
e e
Table 4.3. D and R calculated for MHe+ by using MP4=full method. 50
e e
Table 4.4. D and R calculated for MHe+ by using QCISD (T) and CCSD(T) 52
e e
method and with cc-pv5z basis set.
Table 4.5. D and R calculated for MHe+ by using B3LYP method and with 53
e e
6-31g** basis set.
Table 4.6. D values calculated for MHe+ by using different methods and 54
e
basis sets
Table 4.7. R values calculated for MHe+ by using different methods and 55
e
basis sets
Table 4.8. Previous studies of dissociation energies and bond lengths for 57
LiHe+
Table 4.9. Previous studies of dissociation energies and bond lengths for 57
NaHe+
Table 4.10. Previous studies of dissociation energies and bond lengths for 57
KHe+
Table 4.11. Comparison of Methods and Basis sets in CPU times 60
Table 4.12. Bound vibrational energy levels for the LiHe+ complexes. 61
Table 4.13. Bound vibrational energy levels for the NaHe+ complexes. 61
Table 4.14. Bound vibrational energy levels for the KHe+ complexes 62
Table 4.15. Mulliken charges on the atoms for CCSD(T) method 62

v
LIST OF FIGURES

Page No
Figure 2.1 : Formation and cooling of helium droplets in a gas expansion….. 13
Figure 2.2 : Laser-depletion apparatus.............................................................. 14
Figure 2.3 : Experimental setup for doping metal ions ……............................ 16
Figure 4.1 : PECs of LiHe+ by using MP2 method with pvXZ basis sets.……. 41
Figure 4.2 : PECs of NaHe+ by using MP2 method with pvXZ basis sets……. 42
Figure 4.3 : PEC of KHe+ by using MP2 method with 6-31g** basis sets.… 43
Figure 4.4 : PECs of LiHe+ by using MP2=full method with pvXZ basis sets.. 45
Figure 4.5 : PECs of NaHe+ by using MP2=full method with pvXZ basis sets 46
Figure 4.6 : PEC of KHe+ by using MP2=full method with 6-31g** basis set. 46
Figure 4.7 : PECs of LiHe+ by using MP4 method with pvXZ basis sets.…… 48
Figure 4.8 : PECs of NaHe+ by using MP4 method with pvXZ basis sets.…... 48
Figure 4.9 : PEC of KHe+ by using MP4 method with 6-31g** basis sets…... 49
Figure 4.10 : QCISD(T) method PECs for LiHe+ NaHe+ with pv5z,KHe+ with 51
6-31g**
Figure 4.11 : CCSD(T) method PECs for LiHe+, NaHe+ with pv5z,KHe+ with 52
6-311g(3df,3dp)
Figure 4.12 : B3LYP PECs for LiHe+, NaHe+ and KHe+ with 6-31g**………. 53
Figure 4.13 : PECs for LiHe+ by all methods and basis sets…………………… 58
Figure 4.14 : PECs for NaHe+by all methods and basis sets…………………… 58
Figure 4.15 : PECs for KHe+ by all methods and basis sets……………………. 59
vi
LIST OF SYMBOLS

D : Dissociation energy
e
R : Equilibrium bond length
e
H : Hamiltonian
Ψ : Wave function
ε : Eigenvalue
E/N : Electric field to neutral particle number density ratio
M : Effective mass of the electron-containing bubble
-
P (z) : Poisson distribution for pick-up of k particles
k
P : Stagnation pressure
0
P
: Perturbation Operator

S : Overlep matrix
T
: Cluster Operator

i, j, k, l,… : Occupied spin-orbitals
r, s, t, u,... : Virtual spin-orbitals
A, B, C : Becke constants

vii
SUMMARY
Laser ablation of in situ metals has recently made it possible to immerse a large
number of different metal atoms and ions and small clusters of metal atoms in helium
nanodroplets. Atoms and molecules are readily picked up by He droplets, and their
spectra are detected through the use of either beam depletion following absorption or
laser-induced fluorescence. Within the past three years, a wide variety of molecules,
ranging from metal ions to large organic molecules and a number of van der Waals
complexes and even large metal clusters, have been embedded in He droplets and
studied either in infrared or in the visible region.
The complexes MHe+ (M=Li, Na, K) are of great interest, since they are prototypes
of a closed-shell singly charged cation interacting with a closed-shell atom. From a
more applied point of view, the mobility of metal ions in He is important for the
understanding of chemical plasmas, nucleation phenomena, phase transitions and
structures formed in the atmosphere.
The purpose of the present study is to test the accuracy of the interaction potential
functions for the diatomic complexes, MHe+ (M=Li, Na, K), by using MPn,
QCISD(T), CCSD(T), and DFT methods with highly accurate basis sets. Results for
the potential well depths, zero point energies, dissociation energies, bound
vibrational levels are presented. Some of the methods and basis sets employed here
are computationally too expensive to study further growth of the metal-Helium
clusters (MHe +). From these studies, we determine the most suitable method for
n
studying larger clusters and the level of accuracy of such calculations.
viii