Department of Physics, Chemistry and Biology
Bachelor’s Thesis
Chemical bond analysis in the ten-electron series
Thomas Fransson
LITH-IFM-G-EX–09/2112–SE
Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden
Bachelor’s Thesis LITH-IFM-G-EX–09/2112–SE
Chemical bond analysis in the ten-electron series
Thomas Fransson
Adviser: Patrick Norman
IFM
Examiner: Patrick Norman
IFM
Avdelning, Institution Division, Department Computational Physics
Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden
Datum Date 2009-06-04 Spr˚ak Language Svenska/Swedish Engelska/English ⊠ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats ¨Ovrig rapport ISBN ISRN
Serietitel och serienummer Title of series, numbering
ISSN
URL f¨or elektronisk version
Titel
Title Chemical bond analysis in the ten-electron series
F¨orfattare Author
Thomas Fransson
Sammanfattning Abstract
This thesis presents briefly the application of quantum mechanics on systems of chemical interest, i.e., the field of quantum chemistry and computational chem-istry. The molecules of the ten-electron series, hydrogen fluoride, water, ammonia, methane and neon, are taken as computational examples. Some applications of quantum chemistry are then shown on these systems, with emphasis on the na-ture of the molecular bonds. Conceptual methods of chemistry and theoretical chemistry for these systems are shown to be valid with some restrictions, as these interpretations does not represent physically measurable entities.
The orbitals and orbital energies of neon is studied, the binding van der Waals-interaction resulting in a Ne2 molecule is studied with a theoretical bond length
of 3.23 ˚A and dissociation energy of 81.75 µEh. The equilibrium geometries of
FH, H2O, NH3and CH4are studied and the strength and character of the bonds
involved evaluated using bond order, dipole moment, Mulliken population analysis and L¨owdin population analysis. The concept of electronegativity is studied in the context of electron transfer. Lastly, the barrier of inversion for NH3is studied, with
an obtained barrier height of 8.46 mEhand relatively constant electron transfer.
Nyckelord Keywords
computational chemistry, computational physics, quantum chemistry, bond anal-ysis, electron population, electron transfer
⊠
—
LITH-IFM-G-EX–09/2112–SE
Abstract
This thesis presents briefly the application of quantum mechanics on systems of chemical interest, i.e., the field of quantum chemistry and computational chem-istry. The molecules of the ten-electron series, hydrogen fluoride, water, ammonia, methane and neon, are taken as computational examples. Some applications of quantum chemistry are then shown on these systems, with emphasis on the na-ture of the molecular bonds. Conceptual methods of chemistry and theoretical chemistry for these systems are shown to be valid with some restrictions, as these interpretations does not represent physically measurable entities.
The orbitals and orbital energies of neon is studied, the binding van der Waals-interaction resulting in a Ne2 molecule is studied with a theoretical bond length
of 3.23 ˚A and dissociation energy of 81.75 µEh. The equilibrium geometries of
FH, H2O, NH3 and CH4are studied and the strength and character of the bonds
involved evaluated using bond order, dipole moment, Mulliken population analysis and L¨owdin population analysis. The concept of electronegativity is studied in the context of electron transfer. Lastly, the barrier of inversion for NH3is studied, with
an obtained barrier height of 8.46 mEhand relatively constant electron transfer.
Acknowledgements
First of all, I woulds like to thank my supervisor Patrick Norman for the oppor-tunity of completing this diploma work and for excellent supervision. It has been a great experience. I would also like to thank all the other people of Computa-tional Physics at IFM for the great welcome, the time discussing physics, the time discussing less serious things at breaks, floorball sessions and more.
I wish to thank all my friends for simply being my friends and for weathering through this time where I may not always have been the most present person, thanks.
Last but not least, I wish to thank my family.
Contents
1 Introduction 1
1.1 Quantum chemistry . . . 1
1.2 Relevant elements and molecules . . . 2
2 Electronic structure theory 3 2.1 Many-particle systems . . . 3
2.1.1 The Born–Oppenheimer approximation . . . 4
2.1.2 Antisymmetry . . . 4
2.2 The Hartree–Fock approximation . . . 5
2.3 Basis consideration . . . 5
2.4 Molecular orbitals and applications . . . 6
2.4.1 Construction of molecular orbitals . . . 6
2.4.2 Molecular orbitals . . . 8
2.4.3 Electron density . . . 8
2.4.4 Potential energy surface . . . 10
2.5 Electron correlated methods . . . 10
2.5.1 Post-Hartree–Fock methods . . . 10
2.5.2 Density functional theory . . . 11
2.5.3 Hybrid methods . . . 11
2.6 Molecular properties . . . 11
3 Computational details 13 3.1 Programs used . . . 13
3.2 Basis sets used . . . 13
3.3 Methods used . . . 14
4 Results and discussion 15 4.1 Neon monomer . . . 15
4.1.1 Orbitals . . . 15
4.1.2 Electron density . . . 15
4.1.3 Orbital energies . . . 16
4.2 Neon dimer . . . 19
4.2.1 Basis set superposition error . . . 19
4.2.2 Hartree–Fock . . . 20
4.2.3 Second-order Møller–Plesset . . . 20 ix
x Contents
4.2.4 Perturbation theory . . . 20
4.2.5 Results . . . 20
4.3 Molecular equilibrium geometries . . . 22
4.4 Orbital analysis . . . 22
4.4.1 Hybridization . . . 23
4.4.2 Natural atomic orbitals . . . 23
4.5 Bond order . . . 25
4.6 Bond character . . . 26
4.6.1 Electronegativity . . . 27
4.6.2 Dipole moment . . . 27
4.6.3 Mulliken population analysis . . . 28
4.6.4 L¨owdin population analysis . . . 28
4.6.5 Results . . . 29 4.7 Inversion of ammonia . . . 30 4.7.1 Potential barrier . . . 31 4.7.2 Electron distribution . . . 33 5 Conclusions 35 Bibliography 37
Chapter 1
Introduction
1.1
Quantum chemistry
The branch of physics or theoretical chemistry known as quantum chemistry at-tempts to, as far as possible, explain chemical phenomenas by physical laws. This is done considering the laws of quantum mechanics, which are most prominent on a small scale, such as the atomic scale, even though the govern all scales of interactions.
To fully evaluate the behaviour of matter at the intramolecular level, a treat-ment utilizing non-relativistic quantum mechanics will prove insufficient. This is due to the fact that the electrons in the vicinity of nuclei (especially for the heav-ier elements), acts by such a strong potential that only a relativistic treatment is correct.
This thesis has not included relativistic corrections apart from a study of the ionization energies of neon. The relativistic corrections will in most cases act as a perturbation and can be treated as such, it will not greatly alter any of the presented results.
The present works has also disregarded any strong, weak or gravitational in-teractions. Not including gravitational interaction is valid as its strength relative to electromagnetic interaction is several tens of order weaker. Strong and weak interactions, however, are more of the scale of electromagnetic interactions, but the reach is very small (10−15 and 10−18 meter, respectively) and as molecular
analysis works at a scale of about 10−10 m, it will not interfere [1, 2].
Quantum chemical calculations and theories have a variety of uses in modern science. Depending on the methods used, examples of applications are:
• Calculation of equilibrium geometries and energies [3, 4, 5]. Accuracy may surpass experiments [6].
• Study of chemical reactions and formation of molecule [7, 8], including reac-tions too unstable for experiment [9].
2 Introduction • Determination of molecular properties and construction of molecules with
desired properties [10, 11].
This project have utilized ab initio methods, where no prior knowledge of the system of interest is assumed, and semi-empirical methods where some prior knowledge from ab initio calculations or experiment is assumed. Especially the
ab initio methods are computationally very demanding, but also semi-empirical methods that consider the nuclei and electrons can result in to large calculations for macromolecules, e.g. proteins. Due to this there also exists methods that consider the molecular systems to be composed of atoms (“balls”) and molecular bonds (“sticks”). Electrons are not explicitly considered. These methods, such as molecular mechanics and molecular dynamics, are less demanding for larger systems and can thus be very useful in simulations of e.g. protein folding.
1.2
Relevant elements and molecules
Table 1.1. Atoms treated. Atomic mass given in atomic mass units, u.
Element Symbol Configuration Atomic mass Electronegativity
Hydrogen H 1s 1.008 2.1 Carbon C 1s22s22p2 12.011 2.5 Nitrogen N 1s22s22p3 14.007 3.0 Oxygen O 1s22s22p4 15.999 3.5 Fluorine F 1s22s22p5 18.998 4.0 Neon Ne 1s22s22p6 20.180
-Values are obtained from [2]. Electronegativity according to Pauling [12].
The elements and molecules considered in the present work are listed in Table 1.1 and Table 1.2, respectively. Observe that hydrogen fluoride is given the formula FH in order to avoid confusion with Hartree–Fock.
Table 1.2. Molecules treated. Density in kg/m3
and temperature in K.
Formula Name Density Melting pointa Boiling pointa
Ne Neon 0.900c 24.19 27.24 Ne2 Neon dimer - - -FHd Hydrogen fluoride 990 190.1 292.5 H20 Water 998b 273.2 373.1 NH3 Ammonia 0.77c 195.5 239.7 CH4 Methane 0.72c 89 111.7
Values obtained from [2]. aat 105 Pa. bat 293 K and 105 Pa. cat 273 K and 105
Chapter 2
Electronic structure theory
The main problem in quantum chemistry is that it deals with larges system, com-posed of many (charged) particles under the influence of each other (mainly due to Coulumb interactions and the Pauli principle) and subject to the laws of quantum mechanics.
As this project will not deal with any dynamics, i.e. entities varying in time, and the relativistic effect are as previously stated disregarded, the behaviour of the systems of interest are governed by the time-independent Schr¨odinger equation for a many-particle wave function
ˆ
H|Ψi = E|Ψi (2.1)
where ˆH is the Hamiltonian, Ψ in this case a time-independent wave function and E is the obtained observable: the energy. Further, quantum mechanics states [1, 14], that the probability of finding any particle of interest at a specific point d¯ris
ρ(¯r) = |Ψ(¯r)|2d¯r (2.2)
thus yielding another observable, the probability density of particles.
Consider a water molecule. The molecule is composed of three nuclei, one oxygen nucleus and two hydrogen nuclei, and ten electrons. These can to a good approximation be considered point-like with only an electric charge. This system will then have 3 positive nuclei and 10 electrons, all under the influence of each other. This 13-body system is not possible to consider analytically, so approxima-tions and numerical methods must be applied. This is indeed valid for systems save those consisting of one electron and one nuclei, for instance H, He− etc.
2.1
Many-particle systems
The Hamiltonian in Equation 2.1 for molecular systems is in atomic units (units convenient when considering electronic structure [15, 16]) given by
ˆ H = − N X i=1 1 2∇ 2 i− M X A=1 1 2MA∇ 2 A− N X i=1 M X A=1 ZA riA + N X i=1 X j>i 1 rij + M X A=1 X B>A ZAZB RAB (2.3) 3
4 Electronic structure theory
where the first two sums corresponds to the kinetic energies, the third part to the Coulumb attraction between the M nuclei and N electrons, and the last two the Coulomb repulsion in the electron-electron and nucleus-nucleus interaction.
2.1.1
The Born–Oppenheimer approximation
A first approximation that is generally made and of central importance to quan-tum chemistry, is to separate the movement of the electrons and the nuclei. The electrons are considered to be moving in the influence of each other, and the field generated by the nuclei. By finding the energy optimum of different nuclei config-urations, a large number of molecular properties can be considered.
This separation is known as the Born–Oppenheimer approximation and is a rather good approximation that relies on the fact that a protons and neutrons are more than 1835 times heavier than an electron [2]. The movement of the nuclei is thus far slower then that of the electrons and the electrons can thus be considered separately. Doing so, Equation 2.3 loses the second and last term and becomes the electronic Hamiltonian,
ˆ H = − N X i=1 1 2∇ 2 i − N X i=1 M X A=1 ZA riA + N X i=1 X j>i 1 rij (2.4)
2.1.2
Antisymmetry
The approximate wave function describing a single electron is known as an (spin) orbital and is a function of spatial position and spin,
χ(¯x) with x¯= (¯r, ω) (2.5)
where ¯ris the spatial coordinates and ω is the spin function.
A common procedure is to divide each orbital into a spatial- and a spin-dependent part, χ(¯x) = Ψ(¯r) · α(ω) or Ψ(~r) · β(ω) (2.6) where the spatial (molecular or atomic) orbitals are assumed to form an
orthonor-mal basis
hΨi(¯r)|Ψj(¯r)i = δij (2.7)
and the spin functions are assumed to follow
hα|αi = 1; hβ|βi = 1; hα|βi = 0; hβ|αi = 0 (2.8) In other representations, the alpha spin may be referred to as “spin-up”, while the beta spin is called “spin-down”.
Further, the Pauli principle gives that no two electrons can occupy the same spin orbital, χ. As can be seen in Equation 2.6, this means that two electrons may have identical spatial orbitals Ψ, but then differ in the spin function. For
2.2 The Hartree–Fock approximation 5
many-electron wave functions this implies that the wave functions must satisfy the antisymmetric principle, e.g.
Ψ(¯x1, . . . ,x¯i, . . . ,x¯j, . . . ,x¯N) = −Ψ(¯x1, . . . ,x¯j, . . . ,x¯i, . . . ,x¯N) (2.9)
A mathematical function that follows this antisymmetric property is a determi-nants, as interchanging two rows will change the sign of the determinant product. Thus a generalized N -electron wave function can be formed by using determinants, as follows for the Slater determinant, SD,
Ψ(¯x1,x¯2, . . . ,x¯N) = 1 √ N! χi(¯x1) χj(¯x1) · · · χk(¯x1) χi(¯x2) χj(¯x2) · · · χk(¯x2) .. . ... . .. ... χi(¯xN) χj(¯xN) · · · χk(¯xN) (2.10)
2.2
The Hartree–Fock approximation
The simplest physically acceptable wave function is a single SD, i.e.,
|Ψi = |χ1χ2· · · χNi (2.11)
Using this and the full electronic Hamiltonian given by Equation 2.4, the vari-ational principle [14] thus gives,
E= hΨ| ˆH|Ψi ≥ E0 (2.12)
where E0 is the true ground-state energy for the wave function in consideration.
This is the Hartree–Fock method of finding the best orbitals, and thus involves varying the orbital parameters in such a way that E takes its smallest possible value. This method may also be referred to as the self-consistent field, SCF, method.
2.3
Basis consideration
For the description of the system of interest, a number of basis functions are needed to then be varied for in order to find the energy optimum. For many systems, the orbitals of interest are simply given as a linear combination of some basis functions,
Ψorbitali =
X
ν
cνiφtrialν (2.13)
so that the c are variational parameters to be weighted in such a way that a mini-mum is achieved in 2.12. Observe that the system would be completely described if the summation over ν would stretch to completeness; i.e. a complete set of basis functions where utilized. If such is utilized in combination with Hartree–Fock, the energy obtained would be the Hartree–Fock energy, i.e. the optimal energy for
6 Electronic structure theory
the system utilizing Hartree–Fock. A complete system is, however, not possible for computational reasons, so a truncation must be carried out.
Such a truncation and thus the choice of suitable basis set for a sufficient de-scription of the system, is a matter of properties of interest, system in consideration and computational difficulties. A way to proceed is to use known atomic orbitals, AOs, centered on the participating atoms, as trial wave functions. This may give the molecular orbitals, MOs, by a linear combination of atomic orbitals, LCAO,
ΨMOi =
X
ν
cνiφAOν (2.14)
where the set of cνiare now called the MO-coefficients. Henceforth, the superscript
will be omitted for clarity, and molecular orbitals will thus be written as Ψ while atomic orbitals will be written as φ. Further, discussions of LCAO will imply that the atomic orbitals are really the trial orbitals.
The atomic orbitals used are usually of any of the sorts
χ∼ ǫ−ξr (Slater) and χ ∼ ǫ−ξr2 (Gaussian) (2.15) where the Slater-type orbitals (STO) are physically more correct, but computa-tionally more demanding. The Gaussian-type (GTO) are thus more commonly utilized.
2.4
Molecular orbitals and applications
This consideration of constructing and interpreting molecular orbitals will assume a closed-shell molecule with N electrons, the orbitals formed by a LCAO and constructing a many-particle wave function in the form of a single determinant.
2.4.1
Construction of molecular orbitals
As can be seen in the electronic Hamiltonian, 2.4, the only operators of interest are one- and two-electron operators. For simplicity, quantum chemistry label those as follows (a more complete description can be found in [16, 15] etc.)
ˆ O1= N X i=1 ˆ h(i) and Oˆ2= N X i=1 N X j=1 ˆ V(i, j) (2.16)
If the operators are spin-independent, as in non-relativistic considerations, the operations of such yields
hΨ| ˆO1|Ψi = 2 N
X
k=1
hkk (2.17)
resulting in one-electron integrals, hkk, and
hΨ| ˆO2|Ψi = 1 2 N/2 X i,j (2hΨiΨj| ˆV|ΨiΨji − hΨiΨj| ˆV|ΨjΨii = 1 2 N/2 X i,j 2Jij− Kij (2.18)
2.4 Molecular orbitals and applications 7
where Jij is known as the Coulomb integral and Kij is known as the exchange
integral. It is then convenient to express corresponding operators for these integrals ˆ Jjχi(¯x1) = [ Z |χj(¯x2)|2 |¯r1− ¯r2| d¯x2]χi(¯x1) (2.19) ˆ Kjχi(¯x1) = [ Z χ∗ j(¯x2)χi(¯x1) |¯r1− ¯r2| d¯x2]χj(¯x1) (2.20) so that Jij = hχi| ˆJj|χii (2.21) Kij = hχi| ˆKj|χii (2.22) (2.23) Using operators from 2.16, 2.19 and 2.20, the Fock operator can be formed, as
ˆ f = ˆh+ N X j ( ˆJj− ˆKj) (2.24)
It can now be shown [16], that the spin orbitals acquired by HF are those given by the lowest eigenvalues in the Hartree-Fock equation,
ˆ f|χii = N X j ǫji|χji (2.25)
By choosing a unitary transformation that diagonalizes ǫ, a set of orbitals known as the canonical molecular orbitals can be found. As unitary transformations does not change eigenvalues, which in this case are the energies, it is physically legitimate. Note again that orbitals have no direct physical meaning, so there is no unique set of orbitals for a molecular system.The probability density as given by Equation 2.2 is an observable and thus unique. The canonical orbitals give a simpler form of Equation 2.25, as
ˆ
f|χii = ǫi|χii (2.26)
and are unique for a system with non-degenerate eigenvalues ǫ. These will generally not be localized, but will adopt certain symmetries of the molecule and the Fock operator and will thus be computationally well suited. From these, there can be constructed an infinite number of equivalent orbitals that may be more suiting for interpretations, this will be done in Section 4.4.
Further, for a canonical orbital it can be shown [15], that the eigenvalues of the Fock operator corresponds to ionization energies, as following Koopmans’
Theo-rem. The opposite, electron affinity, is true when considering virtual (unoccupied) orbitals,
EN −1− EN = −ǫa and EN− EN +1= −ǫv (2.27)
where ǫa and ǫv are the orbital eigenvalues for the occupied and virtual orbital,
8 Electronic structure theory
2.4.2
Molecular orbitals
When calculating the molecular orbitals for a molecular system, what is obtained is a rather delocalized set of orthonormal orbitals that are not easily interpreted in terms associated with the classical chemists view of chemical reactions. Strictly speaking, the molecular orbitals obtained does not have a center at any one atom or bond, so interpretations in terms of atomic charges etc. cannot be done.
The construction of molecular orbitals are given in Equation 2.14, and can be expressed in terms of matrix operations are
Ψi=
X
ν
cνiφν ⇔ Ψ = φTc (2.28)
where c is a matrix formed by the MO-coefficients.
While the molecular orbitals are orthonormal (as by Equation 2.7), orthonor-mality or even orthogonality is typically not achieved for the atomic orbitals used, but the following is true
hΨi|Ψji =
X
α,β
c∗αicβjhφi|φji = δij (2.29)
where the inner product between two arbitrary atomic orbitals is called the overlap
matrix, S, as
Sαβ= hφα|φβi (2.30)
As can be seen, atomic orbitals are orthonormal only if the overlap matrix is the identity matrix. The overlap matrix is also Hermitian,
Sαβ= hφα|φβi = hφβ|φαi∗= Sβα∗ (2.31)
By the criteria that the molecular orbitals are orthonormalized, it follows that the overlap matrix has ones in the diagonal and all off-diagonal elements have values with an absolute value less than or equal to unity.
It can be seen by considering Equation 2.29, that the following is generally true
cc†6= E but cSc† = E (2.32)
Further, symmetrically orthogonalized atomic orbitals may be constructed from the atomic orbitals simply by considering
φ′= S−1/2φ (2.33)
2.4.3
Electron density
The probability density of electrons, or the electron density, is given as the square modulus of the orbitals, as in Equation 2.2. Starting from this, it may of interest to obtain the probability of finding k electrons in a volume element d¯r, regardless of the positions of the others and the electrons spin. While considering this, it may be of interest to consider the kernel
γk(¯r1, ..,r¯k,r¯′1, ..,r¯′k) =
Z
2.4 Molecular orbitals and applications 9
this is known as the k:th order reduced density function.
With k = 1 the first-order reduced density function is given. Since all electrons are indistinguishable, it has the form
γ(¯r1,¯r1′) = N
Z
Ψ(¯x1, ..,x¯N)Ψ∗(¯x′1, ..,x¯′N)dω1d¯x2..d¯xN (2.35)
The interpretation of this function is such that the diagonal entries gives the electron density function, i.e. the probability of finding an electron at position ¯r1.
The integral of this density function thus equals the total number of electrons. ρ1(¯r1) = γ(¯r1,r¯1) = N Z Ψ(¯x1, ..,x¯N)Ψ∗(¯x1, ..,x¯N)dω1d¯r2..d¯rN (2.36) N = Z ρ1(¯r1)d¯r1 (2.37)
Now, consider a a closed-shell molecule described by a single determinant, i.e. HF. With no difference made for different spin, this will yield an occupation of two for the occupied MOs. With the ground state Hartree–Fock wave function, the one-electron density function may give the electron density function as
ρHF(¯r) = γHF(¯r,¯r) = 2 occ X i Ψi(¯r)Ψ∗i(¯r) = 2 occ X i X α cαiφα(¯r) X β c∗βiφ∗β(¯r) (2.38) =X αβ [2 occ X i cαic∗βi]φα(¯r)φβ(¯r) = X αβ DαβAOφα(¯r)φβ(¯r)
where DAO is the density matrix in the basis of atomic orbitals. The density
matrix specifies the electron density in the given basis. In combination with the overlap matrix it thus specifies the electron density for the system of interest. It is thus easy to see that the following gives the total number of electrons N ,
N = 2 occ X i Z |Ψi(¯r)|2d¯r= X α,β DαβSβα (2.39)
where, for simplicity, we have denoted DAO= D.
It may also be constructed a density matrix in MO-basis, this should follow (remember that this is a closed-shell molecule)
DMO=
2δij j≤ occ
0 otherwise (2.40)
Following Equation 2.28, the two density matrices can be connected by
D= cDMOc† (2.41)
and, if considering Equation 2.32
10 Electronic structure theory
2.4.4
Potential energy surface
Having obtained the molecular orbitals for an arrangement of the nuclei, the en-ergy associated with this can be determined. By computing the enen-ergy for many such arrangements (configurations), the potential energy surface, PES, can be constructed. As potential minima represents stable configuration, the search for equilibrium geometries is a search for minima in the PES.
2.5
Electron correlated methods
An insufficiency of Hartree–Fock is the inability the approximation to properly treat electron correlation. Electron correlation corresponds to the correlation of electron movement due to Coulomb repulsion, and as HF treat electron-to-electron interaction with an average interaction, it will yield a behaviour of the electrons to be closer to each other than what is the case. There are a number of methods to better evaluate the proper behaviour, some are given below [16].
2.5.1
Post-Hartree–Fock methods
A method to proceed by, is to take Hartree–Fock as a start and attempt to improve the result by addressing electron correlation in some manner. As this may be done without knowledge of specific systems, i.e. no necessary parametrization, it may still be examples of ab initio methods. It can then be of interest to find the correction energy, the energy HF is to be corrected by in order to approach the better energy obtained by this improved method. This would be given simply as
Ecorr= E0− EHF (2.43)
where EHF is the Hartree-Fock limit energy. Th energy of the better method is
denoted E0, and is generally not the exact energy of the system. Observe that
the HF energy is always larger than the correct energy, see Equation 2.12, so the correction energy should be negative, otherwise the new method is less accurate than HF.
A few examples of these post-Hartree–Fock methods are given here, there are more available [16]:
• CI; Configuration interaction uses a linear combination of Slater Determi-nants and the variational principle to find the energy minima. In the limit of full basis set, this would yield the physically correct energy. It can be computationally challenging and does not properly treat molecules in large separation compared to single molecules, it is not size-consistent [15]. • MCSCF; Multi-configurational self-consistent field uses a linear
combina-tion of “suitable” SD and vary the MOs and the MO coefficients, c, at the same time. It is thus simply an improvement of SCF that can be expected to yield a more qualitatively correct description [15, 17].
2.6 Molecular properties 11
• Møller-Plesset; Møller–Plesset perturbation theory adds a perturbation on the Fock operator to account for the correlation energy. Depending on the order of energy correction, the approach is given an integer, MP1, MP2, MP3 etc. MP1 is the first-order correction and yields the Hartree–Fock energy. While not being variational nor able to correctly account for static correlation energy, Møller–Plesset gives good results even for small corrections such as MP2, and it is size-consistent [16].
2.5.2
Density functional theory
Density functional theory, DFT, uses the electron density, ρ, rather than the electron wave functions, to compute electronic energies. The theory consider
func-tionals, i.e. functions of functions, as the energy depends on the electron density, which in turn depends on the electron wave functions. The functionals are however not generally known, and require some parametrization either from experiment or
ab initio calculations, to ensure proper quality. DFT still offers good results at low computational costs [16].
2.5.3
Hybrid methods
A method of improving DFT is to take into account exact exchange energies given by Hartree–Fock theory. Such mixing of theory is known as hybrid methods and may give very good results at a low computational cost [18].
2.6
Molecular properties
In the preceeding sections, only molecules in equilibrium, lacking any external fields have been treated. However, when a field of some sort, may it be electric, magnetic etc, is applied on a molecule, it will change the energy and possibly the geometry of the molecule.
The applied fields will in most cases contribute with small interactive energies and can thus be considered as an perturbation. The treatment of perturbation in quantum mechanics is an area with large fields of application, and will not be treated rigorously, see [1, 14]. What is important is that any perturbation on a system can be treated by Maclaurin expansions, so the energy of a molecule perturbed is E(F, B, ...) = E0+ ∂E ∂FF+ ∂E ∂BB+ . . . (2.44)
where E0 is the energy of the isolated system and F , B correspond to applied
electronic field and magnetic field, respectively. Other perturbing fields are pos-sible but will not be considered. E0 is the unperturbed energy. This is valid as
even macroscopical large entities, say a magnetic field of 10 T, will not be large in terms of molecular entities, the field equaling 4.5 · 10−5 atomic units.
Molecular propertiesare defined as the partial derivatives in 2.44 given at zero field strength. Further, the properties will be static if the perturbing fields are
12 Electronic structure theory
time-independent, and dynamic if the fields are time-dependent. Static properties will be treated henceforth.
Considering some first-order properties, they are defined as follows m0i = − ∂E ∂Bi F,B,···=0
permanent magnetic dipole moment (2.45)
µ0i = − ∂E ∂Fi F,B,···=0
permanent electric dipole moment (2.46)
etc.
Consider a perturbative field and the resulting first-order molecular property. A variational wave function with parameters λ may be constructed in this field of perturbation. The energy such an variational wave function is a function of the field and the variational parameters, designating this ε(A, λ). Varying the parameters, the optimal wave function and energy can be obtained for the system of interest, as
E(E, λ(A)) = min
λ ε(A, λ) (2.47)
for some field A, with the optimization condition ∂E
∂λ = 0 (2.48)
Together with Equation 2.44, the following can thus be seen, dE dA = ∂E ∂A + ∂E ∂λ ∂λ ∂A = hΨ| ∂ ˆH ∂A|Ψi (2.49)
Chapter 3
Computational details
3.1
Programs used
The calculations have mostly been carried out with the non-relativistic program Dalton [20]. Section 4.6 and 4.4, however, utilized Gaussian 03 [21] for the cal-culations, as this was more fitting for the tasks at hand. Finally, the relativistic calculations in Section 4.1 were done with Dirac [22].
3.2
Basis sets used
The minimal basis set STO-3G has also been utilized for illustrations of hybridiza-tion of methane and obtaining the electron density of neon. It has a minimal number of basis functions, with only as many basis functions as there are occupied AOs. This basis set is not particularly good at obtaining accurate results due to the small size, but it retains the essential behaviours of chemical systems and can thus be used for interpretations. As it is the minimal number of orbitals corre-sponding to the ground-state orbitals that is taken into account, it illustrates the behaviour of the occupied orbitals sufficiently for the tasks it is utilized on in this work.
Most calculations in this project have been utilizing Dunning’s basis set cc-pVTZ, correlation consistent, polarized Valence Triple Zeta. It has three times the minimal basis for valence electrons, minimal basis for the core and adds polarizing functions. This is a relatively good basis set that is fitted for recovering the correlation energy of the valence electrons, although it is not good at correctly evaluating long-range behaviour.
Some calculations have been computed with the smaller set cc-pVDZ, which adds less polarization functions and has two times the minimal basis for the valence electrons. It is not expected to be as accurate as cc-pVTZ, although useful for some illustrations.
Some evaluations of neon has utilized the basis set taug-cc-pVTZ, an example of cc-pVTZ with added diffuse functions in order to better describe behaviour at a
14 Computational details
longer distance from the atoms they are centered upon, i.e. long-range behaviour.
Table 3.1.Basis sets used and atomic orbitals utilized by those. The left side indicates the number of primitive Gaussian orbitals, and the right side the number of contracted.
Atom Basis set Atomic orbitals Reference
H STO-3G [3s|1s] [23, 15] cc-pVDZ [4s1p|2s1p] [24, 25, 26] cc-pVTZ [5s2p1d|3s2p1d] [24, 25, 26] C,N,O,Ne STO-3G [6s3p|2s1p] [23, 15] cc-pVDZ [9s4p1d|3s2p1d] [24, 25, 26] cc-pVTZ [10s5p2d1f |4s3p2d1f] [24, 25, 26] taug-cc-pVTZ [13s8p5d4f |7s6p5d4f] [27]
3.3
Methods used
The method most commonly used for the project is B3LYP, a combination of
Becke’s three-parameter exchange functional, [28] , and Lee-Yang-Parr correlation
functional [29]. B3LYP is a DFT method with hybrid functionals that provides qualitative results at a lower cost than ab initio methods of the same accuracy.
However, B3LYP fails to properly account for weak interactions such as van der Waals interaction. For this reason, a post-HF method, MP2 will be utilized in Section 4.2.
Further, some sections utilize Hartree–Fock in order to obtain the canonical orbitals and as an illustration of the results by the use of different methods.
Chapter 4
Results and discussion
4.1
Neon monomer
As stated in Section 2, only the hydrogen atom and other ionized atoms, composed of only two particles, can be evaluated analytically. In this section a neon atom (neon monomer) is studied with approximate methods and interpretations of the results are given.
4.1.1
Orbitals
First, the electron wave functions, orbitals, of the atom are calculated and the shape of those plotted. Only the occupied orbitals in ground state are considered. Those are expected to have shapes corresponding to the spherical harmonics de-scribed in fundamental quantum mechanics [14, 1], and this is seen to be valid in the resulting isoelectric surfaces, Figure 4.1.
The isoelectric surface is interpreted as the surface of point where the square value of the wave function, i.e. the probability, has the value that is sought for. In this case, the surfaces with the isoeletric value 0.2 are sought for, as
|Ψ(¯r)|2= 0.2 (4.1)
4.1.2
Electron density
As previously stated, orbitals (atomic or molecular) are not observables. As such, they are not measurable, but can give observables and be utilized as tools for analysis of the systems of interest. An observable that can be obtained is the electron density from Equation 2.2, and thus any valid orbitals must yield this density. Also note that it is the total electron density that is the observable, there is no easy way to experimentally distinguish between different electron wave functions.
16 Results and discussion 1s 2s 2px 2py 2pz
Figure 4.1. Occupied ground-state orbitals of Neon. The surface represent the isoelec-tric value 0.2.
Using the different orbitals given at HF/STO-3G level, the electron density can be given as a function of radial distance from the neon nucleus as in Figure 4.2.
In this Figure, it can be seen that the 1s-orbital yields a large electron density near the nuclei that fades quickly, its total contribution corresponding to the charge of two electrons. The 2s- and 2p-orbitals yield a large electron density at a larger distance, their contribution corresponding to the charge of two and six electrons, respectively, as expected.
Studying the sum of electron densities, it can be seen that there is an increased probability of finding electrons in two different “shells” of increased probability. This is in rather good agreement of the chemical view of electron shells determined by the principal quantum number n. Further, Figure 4.2 also include the van der
Waals radiusfor neon. van der Waals radius is a measure of the atomic size (since only the probability for the electrons can be given, there is no definite size), which is defined as the the half-distance between two nuclei of two atoms of neighbouring molecules [31]. The van der Waals radius seems to be reasonably positioned, with small possible orbital overlap beyond.
4.1.3
Orbital energies
Considering Koopmans’ theorem, Equation 2.27, the ionization energies of canon-ical orbitals are simply the eigenvalues. Using this, it is possible to calculate these energies at the HF level of theory. This is done both for the relativistic and
4.1 Neon monomer 17 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 Radial distance, (Ã) Electron density, ρ 1s 2s (lower curve) 2p Full density
Figure 4.2. Neon electron density as a function of radial distance from neon nuclei. Included is the van der Waals radius, from [30]. Obtained using HF/STO-3G.
non-relativistic cases, and the results are given in Figure 4.3.
As can be seen in this figure, there is a splitting of 2p orbitals as well as there is a lowering of the orbital energies for the relativistic calculation, as expected [1, 14].
The splitting of energy levels can be contributed to the spin-orbit effect that in relativistic quantum mechanics also couples the total angular quantum number
j, equaling
j= l + s or j= |l − s| (4.2)
with the orbital energies. Further, there is a general lowering in orbital potential due to length contraction, as the electrons experience contraction of the distance to the nuclei and will thus be more strongly bond.
As can be seen in this example, relativistic corrections must be included for an accurate description. The correction is, however, not so large that it alters the general behaviour of the atom, nor is the energy correction big in a relative consideration (being at most about 1%). Further, as neon is the heaviest element in consideration, the relativistic effects will be smaller for the other elements. It is thus reasonable to disregard of this effects in the present work. Observe that any consideration with heavier elements will yield a greater inaccuracy as the relativistic effects are larger for larger atoms.
18 Results and discussion −34 −33 −32 −31 −4 −3 −2 −1 0 1s (2) 1s (2) (2) (2) (2) 2s 2s 2p (6) 2p1/2 2p3/2 (4) −32.77240 −1.930400 −0.850400 −32.815654 −1.932765 −0.847849 −0.8434809 Energy, E h
Figure 4.3. Ionization energies obtained with non-relativistic HF/cc-pVTZ for the left side, and relativistic DHF/cc-pVTZ calculations for the right side.
4.2 Neon dimer 19
4.2
Neon dimer
As explained in Section 2.5, electron correlation must be added to any precise description of molecular systems. In fact, for covalent bonds, intermolecular in-teractions are rarely a consequence of orbitals spanned in a network of molecules, [31], they can rather be considered as having closed orbitals, just as in the case of monoatomic noble gas. If only the assumption made by HF is considered, this would mean that no intermolecular forces between such molecules will exist, and only a non-reactive gas can be the result of such a system, no solids or liquids.
In this section, the intermolecular interactions arising from electrons correlating their movement, thus creating induce dipole moments, will be considered on a neon dimer. The forces arising from this is known as van der Waals forces and the resulting bond as a van der Waals bond (this is sometimes known as London dispersion forces, as well). Except for extreme cases of ionization, this is the only interaction neon and the other noble gases can experience, and is thus of interest when considering such systems. van der Waals interactions can also be of interest when considering intermolecular systems for above mentioned molecules, for its influence in protein folding etc.
This system will thus be studied using a variety of methods, illustrating the validity of those. In this calculations, B3LYP is not utilized due to its inability to properly describe weakly bound systems [32]. Generally, functionals in DFT are not capable of properly account for dispersion interactions [16].
4.2.1
Basis set superposition error
When attempting to compute a PES for a neon dimer, the most intuitive procedure is simply to take the difference in energy for the neon monomer and a dimer with different interatomic distances. Performing such a calculation would however not yield a correct result.
The reason for this failure is the basis set superposition error, BSSE. This arise when a system and its components are calculated and energies obtained compared. It is due to the fact that in any finite basis calculation, the basis (if not already corrected) used will be larger for the full system [16, 33].
Consider the neon dimer. When both neon atoms are considered at the same time, they will use parts of the basis functions centered at each other, thus giving a better description of the atoms and lowering energies when compared to a single atom. This will in turn mean that any comparison between the energies of the two systems will be incorrect, overestimating interaction energies [33].
An intuitive procedure of correcting this fault is the counterpoise correction
method, CP [16]. CP estimated the BSSE as the difference between the differ-ent compounds energies in their own basis sets with the energy of the differdiffer-ent compound in the basis set of the full complex, both with the geometry from the complex. The energy of the compound in the full basis set is given by locating the basis functions for all compounds in space, thus giving so called ghost orbitals. This method is only approximate, and its validity is the subject of discussion [34].
20 Results and discussion
4.2.2
Hartree–Fock
Using Hartree–Fock methods on a neon dimer is not expected to yield any bonding at all. This is due to that HF consider electron-electron interaction as the interac-tion of an electron with an effective charge cloud, thus preventing any correlainterac-tion. The calculation is carried out as an illustration.
4.2.3
Second-order Møller–Plesset
In order to properly evaluate the van der Waals interaction, it is necessary to use some improved method for the electron correlation. For this reason, MP2 will be utilized and is expected to at the very least yield a binding minima.
4.2.4
Perturbation theory
When considering a neon dimer with such a large interatomic separation, it is reasonable to assume that the correlation interaction is relatively small when com-pared to total energies etc. Assuming this, the correlation can be considered as a perturbation, and can be evaluated using perturbation theory, see [14] for details. The precise method of obtaining this perturbation will not be given here, it can, however, be shown to equal [35],
EvdWNeNe= − 3~ πR6 ∞ Z 0 [¯αNe(iω)]2dω= −C6 R6 (4.3)
The value of C6 was found using HF/taug-cc-pVTZ, with a numerical value of
5.379 a.u. Observe that this method is only valid in the region where negligible electron overlap occurs, the van der Waals-region. Further, note that this value is obtained by HF, a computationally less challenging method than MP2.
4.2.5
Results
Using these different methods, the potential energy surface was calculated accord-ing to Section 2.4.4. This is easy in this case, as there are only two atoms in consideration, so only the distance is varied. The result can be seen in Figure 4.4, considering interatomic distances of 4–7.5 a.u. (≈ 2–4 ˚A). Inserted in this graph is also the BSSE for the MP2 plot, evaluated with CP, and a vertical line illustrating the potential minima obtained by MP2.
As can be seen, HF does not give any minimum as it is incapable of evaluating van der Waals interactions. Perturbation theory gives a good result in the van der Waals-region, although it is then important to evaluate the extent of said region. Using the van der Waals-radius as explained in Section 4.1.2, the van der Waals-region would be distances of above 5.82 a.u. (3.08 ˚A). The BSSE for MP2 is considerable, with a value at equilibrium of about −579.7 µEh.
Finally, MP2 gives the best results, with a potential minima of −81.8 µEh at
4.2 Neon dimer 21 4.5 5 5.5 6 6.5 7 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Interatomic distance, a.u.
Energy, mE h MP2 HF PT BSSE
Figure 4.4. The potential energy surface of neon dimer from HF, MP2 and PT using taug-cc-pVTZ. BSSE for MP2 is also included. All quantities are expressed in atomic units.
22 Results and discussion of about −127 µEhand 5.84 a.u. (3.09 ˚A) [32, 36]. As can be seen, the theoretical
values do not make a perfect fit, but this can be explained by the difficulties of evaluating the interaction.
4.3
Molecular equilibrium geometries
As Ne2 is the only molecule of those in consideration that is governed by only a
weak interaction, the others can be considered with B3LYP with no large loss of accuracy.
Using obtained values for the neon dimer and performing the calculations for the other molecules, the equilibrium geometries are calculated and compared to experiment in Table 4.1. Observe that neither the theoretical nor the experimental value of Ne–Ne is very precise, due to the unstable nature of the bond. Neon will not be considered in any further considerations.
Table 4.1. Calculated and experimental geometries for molecules. Bond lengths are given in ˚A and bond angles in degrees.
Molecule Parameter Theory Experiment
Ne2a Ne–Ne 3.23 3.09 FHb H–F–H 0.923 0.917 H2Oc H–O–H 0.962 0.957 HOH 104.5 104.5 NH3c H–N–H 1.014 1.017 HNH 106.5 107.8 CH4c H–C–H 1.089 1.094 HCH 109.5 109.5d
aobtained from MP2/cc-pVTZ according to Section 4.2, experimental value refer
to [32] ([37] have 3.1 ˚A).bobtained from [37]. cobtained from [38]. dvalue obtained
for methane of tetralhedral shape, as is assumed for experimental values.
4.4
Orbital analysis
As stated in Section 2.4.1, the canonical orbitals fulfilling Equation 2.26 are suit-able for computations, but generally delocalized over the entire molecule. This delocalization makes interpretation in terms of chemical bonds hard, and there is little consistency between orbitals for chemically equivalent compound, such as a methyl-groups.
In this section the canonical orbitals given using HF will be transformed by a unitary transformation for easier interpretation. The procedure of performing a non-unitary transformation of orbitals obtained by othe methods will also be discussed. Observe that there exist infinitely such transformations, and it is thus a choice to use the most suitable for the situation at hand.
4.4 Orbital analysis 23
4.4.1
Hybridization
It is first of interest to choose some unitary transformations that construct localized
molecular orbitals, LMOs, from the canonical orbitals. Those should then be constructed in such a way that they solve previously stated issues. These can thus more easily be utilized for futher analysis [39, 40, 41].
The LMOs may be constructed by optimizing the expectation value of some two-electron operator, Ω, hΩi = Nbasis X i=1 hφ′ iφ′i|Ω|φ′iφ′ii (4.4)
The process may be called hybridization, as it can be interpreted as constructing hybrid molecular orbitals from atomic orbitals.
In this calculation, a set of canonical orbitals obtained by HF/STO-3G cal-culation on methane are be treated with the Boys localization scheme [42, 16], where the expectation value to be minimized is that of the square of the distance between to orbitals, thus giving LMOs with minimal spatial extent, as
hΩiBoys= Nbasis X i=1 hφ′ iφ′i|(¯r1− ¯r2)|φ′iφ′ii (4.5)
The resulting orbitals before and after the localization are given in Figure 4.5. Isoelectric values for the orbitals are 0.20, except for the top-middle orbital, where it is 0.18. Further note that all obtained orbitals are not given here, as three canonical orbitals with the same form as the top-left thought different orientations, and four LMOs of the form as the bottom-right with different orientations is obtained.
Studying the result, it can easily be seen that the LMOs provides localized MOs, while the canonical MOs are not easily identified as any specific bond. The exception is the two orbitals to the left, as these are the core orbital of carbon, which stays rather unaffected by the formation of methane. The other four LMOs can be interpreted as sp3-orbitals, i.e. hybrids formed by a combination of 1s
from hydrogen, and a sp3 from the carbon. sp3 is a hybrid orbital formed by the
combination of 2s and all 2p:s.
4.4.2
Natural atomic orbitals
Consider the density matrix D (in AO-basis). This matrix is given from Equation 2.4.3, as D= 2 occ X i=1 cic† i = 2 occ X i=1 (cic†i)†= D† (4.6)
and it is thus Hermitian. Observe again that closed-shell molecules are consid-ered. Any Hermitian matrix may be diagonalized in a basis of eigenvectors, and finding those eigenvectors and corresponding eigenvalues is equivalent of finding the occupation numbers and natural orbitals, NO.
24 Results and discussion
Figure 4.5. Canonical MOs and Boys LMOs after hybridization. Obtained using HF/STO-3G.
Originally, those natural orbitals where considered for CI wave functions, where they provide the system of fastest convergence [16, 43]. If utilized on wave functions not obtained by full CI, fastest convergence is not guaranteed, but it still gives a good measure on the importance of a orbital (by the occupation number) and can thus further be utilized for determining the orbitals to be included in a MCSCF wave function.
The natural orbitals are, however, not localized as the LMOs. It may be of interest again to find orbitals that can be localized to a bond, or possibly to individual atoms. Natural atomic orbitals, NAO, can be constructed as the atomic orbitals with maximum occupancies. The result will be orbitals that can be divided to those with large occupancies (natural minimum basis) and those with occupancies close to zero (Rydberg orbitals). Generally, it is desired to have a procedure that retains the size of the natural minimum basis, only increasing the number of Rydberg orbitals as the size of the basis is made larger.
The construction of natural orbitals can be found elsewhere [16, 43, 44], but the principles of this follows.
Consider a system where the basis functions have been ordered in such a way that all atomic orbitals for atom A are ordered before all others, those belonging
4.5 Bond order 25 to B before those to C etc. The density matrix will then look like
D= DAA DAB . . . DBA DDBB . . . .. . ... . .. (4.7)
where DAA is the density matrix block with basis functions belonging only to
A, etc. The natural atomic orbitals for atom i can then be defined as those that diagonalize the Diiblock. The set of all such orbitals for a system will generally not
be orthogonal, so for a physically acceptable transformation, an orthogonalization procedure must be completed.
The orthogonalization of this obtained set of NAOs, labeled pre-NAOs, is an
occupancy weighted procedure, where the natural minimum basis is first made orthogonal, then proceeding with the Rydberg orbitals to ensure minimal distor-tion of the system. It is here important to note that the NAO-procedure is not
a unitary transformation, so it does not guarantee the conservation of charge, if used poorly (a non-unitary transformation does not retain the eigenvalues, as op-posed to the construction of LMOs). It may also improperly evaluate diffuse basis functions, as those may describe an electron density far from the atom they are centered on, but the NAOs will still consider this density on this atom.
The NAOs may then be unitary transformed to natural bond orbitals, NBO, which then considers the off-diagonal blocks Dij and gives localized bond orbitals.
Those bond orbitals may then be interpreted in terms of which atomic orbitals that is involved in a specific bonding. Transformations to orbitals beyond NBO is also possible [45].
Note that both NAO and NBO are uniquely described by the density matrix, thus uniquely described by the wave function Ψ itself [43]. They should thus be found to have the desired properties regardless of the basis used, presuming it is sufficiently constructed to obtained a proper natural minimal basis.
Neither the NAOs nor NBOs has been computed for this thesis, due to com-putational difficulties.
4.5
Bond order
In order to study the strength and location of molecular bonds, it is possible to consider the Bond Order, BO, [46, 47, 48]. This utilizes the density matrix D and overlap matrix S in order to determine the chemical view of bond multiplicity between two atoms, as well as in some degree determining the strength of those bonds. The bond order is defined as
BOAB= Nbasis X α∈A Nbasis X β∈B (DS)αβ(DS)βα (4.8)
The interpretation of this number is such that values close to integers are expected. Those integers will in turn be seen as the number of chemical bonds
26 Results and discussion between the atoms, i.e., 0 equals no bonding, 1 a single bond, 2 a double bond etc. Integer values would be expected for homoatomic molecules, but this will not occur in the following calculations [46]. The deviation can be interpretated as relative bond strength. Further, delocalized bonds as those between carbon atoms in benzene, could have a value of 1.5, representing a delocalized double bond [49]. BO is basis set dependent, so any comparison is only valid in systems where the same basis set is used. The basis set dependence is, however, not as large as for atomic charges [16], as explained in the following section. Further, BO require atom-centered basis functions.
Table 4.2. Bond order strength between different atoms in the molecules. Values obtained with B3LYP and basis set given in the footnote.
Molecule Bond Basis Bond order
FH F–H A 1.01 F–H B 1.05 H2O O–H A 1.02 B 1.03 H–H B 0.01 NH3 N–H A 0.99 B 1.01 H–H B 0.01 CH4 C–H A 0.97 B 1.00 H–H B 0.00 A = cc-pVDZ, B = cc-pVTZ
The bond order is computed for all bonds in the molecules of interest, and the result is given in Table 4.2. This results seem reasonable, as it would mean no bonds between hydrogen atoms, and a single bond H–X (X=F, O, N, C). Further, it may be noted that a study of the bond orders using the same basis set would mean that the bond strengths follow, for cc-pVTZ, F–H > O–H > N–H > C–H, as is expected. For cc-pVDZ O–H > F–H, this could possibly be due to cc-pVDZ being a less accurate basis than cc-pVTZ.
4.6
Bond character
When the molecular bonds are identified, it may be of interest to identify the bond
characterof those. In chemistry, there is usually made a distinction between three main types of bonds with the approximate behaviour as below:
• Covalent bond, a type where electrons are “shared” between two atoms, i.e. the orbitals are centered on several atoms.
• Ionic bond, a type where an electron is said to be transfered between two atoms, yielding ions and localized atomic orbitals.
4.6 Bond character 27 • Metallic bond, a type where electrons are shared between several atoms,
yielding completely delocalized orbitals.
There is no unique way of determining which type a bond belongs to, as the properties varies continuously.
When considering covalent and ionic bonds, which are the types considered in this thesis, we may study the electron transfer between the different atoms. The electron transfer, or charge transfer, is the transfer of electronic charge due to the molecular orbitals shape. If the (bond) molecular orbital has such a shape that the electron density is more localized at one end, i.e. a greater probability of the electron to be in the vicinity of one nucleus, the effect will be such that there is a redistribution of charge in that direction. Thus, if this redistribution would equal one elementary charge, this would be interpreted as the transfer of one electron, yielding two ions with charges ±1.
There are different way of giving a qualitative measure of this, as is shown in this section.
4.6.1
Electronegativity
In chemistry, at least the main elements are given a property called
electronega-tivity. This originates in the theories of Pauling, and the numbers given by him are the most used ones, although others exists [50, 31]. Electronegativity is a qual-itative measure of an element’s ability to attract electrons in a bond, thus giving the electron transfer by considering the difference in electronegativity, as
χA− χB= ∆χ (4.9)
where χi is the electronegativity value given for atom i. Note that the
interpreta-tion is such that the larger values means a greater attracinterpreta-tion.
The character of a bond can thus be obtained by some rather arbitrary defi-nition of bond characters for values of ∆χ. For example, [51] set the distinction between ionic and covalent bond at ∆χ = 1.7.
Note that electronegativity is not a measurable quantity and that it only gives a qualitative judgement of the nature of the bond. Further, functional groups may also be given a value of electronegativity in order to get a more general property, but this is not of interest in this thesis.
4.6.2
Dipole moment
As seen in Section 2.6, first-order molecular properties can be calculated as an expectation value, following 2.49. Thus the electric dipole moment of a molecule can be given by, using Equations 2.46 and 2.49
µ0i = − ∂E ∂Fi F,B,...=0= −hψ0| ∂ ˆH ∂Fi|ψ0i (4.10) Further, this dipole moment can be utilized for studying the electron distribu-tion, if the equation for dipole moments is considered [2].
¯
28 Results and discussion where ¯µis the dipole moment, Q is the electric charge and ¯dis the distance between the charges.
As this permanent dipole moment arise from said redistribution of charge, it represents an observable and thus is measurable. As such, it could be believed to always represent the true electron transfer, but this would not be possible in the example of molecules with some symmetry, for instance methane, for which different redistributions cancel (partially or fully).
4.6.3
Mulliken population analysis
As explained in Section 2.4.3, the density matrix D and the overlap matrix S give the probability of finding an electron in a given volume, as by Equation 2.4.3. It is possible to interpret the number of electrons associated to a specific basis function, thus further yielding the number of electrons associated with a specific atom, by
ρA= Nbasis X α∈A Nbasis X β DαβSαβ= Nbasis X γ∈A (DS)γγ (4.12)
where A is the atom of interest. This analysis is known as Mulliken population
analysis, MPA, and is rather commonly utilized. The net charge of the atom is then simply
qA= ZA− ρA (4.13)
where ZA is simply the charge of the nucleus.
MPA requires the basis functions used to be centered on atoms, but even with that, there exists a number of other issues [16]:
1. The population in an orbital may be < 0 or > 2, which clearly is physically unacceptable.
2. The analysis implies that “shared” electrons (given in the off-diagonal blocks) of the DS product matrix is shared equally between the participating elec-trons. There is no objective reason for doing so.
3. Diffuse basis functions centered at one atom will give occupancy at that atom, even if it illustrates an electron density far away.
4. Dipole, quadrupole, etc., moments are generally not conserved. These issues may partially be solved in the section below.
4.6.4
L¨
owdin population analysis
In order to solve some of the difficulties of MPA, there exists another population analysis that utilize the symmetric orthogonalized atomic orbitals, as given by Equation 2.33. This is indeed valid, as the total number of electrons may indeed be given by any combination following
N = Nbasis X γ (DS)γγ= Nbasis X γ (S1−αDSα)γγ as trAB = trBA (4.14)
4.6 Bond character 29 so in this case L¨owdin population analysis, LPA, is defined as
ρA= Nbasis
X
γ∈A
(S1/2DS1/2)γγ (4.15)
LPA is said [16], to solve the first three issues given in MPA. However, care must be taken if Cartesian basis functions are used [52, 53, 54].
4.6.5
Results
Using the overlap matrices S, the density matrices D, the geometric structure from 4.3 and the dipole moments calculated, the results are given in Table 4.3. The table shows the electron transfer in each bond, or the electronic charges of the hydrogen atoms, in the molecules by the method used. The following behaviours can be deduced from these specific results:
1. The dipole moment approach predicts larger electron transfer than any of the population analysis.
2. LPA gives physically unrealistic populations for cc-pVDZ, with an electron transfer to hydrogen in all cases.
3. MPA gives larger (to the absolute value) electron transfers than LPA, save for CH4 using cc-pVDZ.
4. Electron transfer is smaller (to the absolute value) for the smaller basis in all cases save CH4.
5. Analogously to the results in Section 4.5, the absolute value of electron transfer gives a bond strength ordering as F–H > O–H > N–H > C–H, as expected.
Any population analysis is somewhat arbitrary by nature, and different values obtained by the different methods are thus expected. The different values for the same basis set may be understood from this, but the physically unrealistic results obtained at LPA/cc-pVTZ is not promising.
As explained elsewhere [52, 53], a L¨owdin analysis is not rotationally invariant if the basis functions used are non-orthogonalized Cartesian basis functions. How-ever, the present calculations utilize basis sets that uses spherical basis functions, so this explanation is not valid. It is further of interest to note that the values obtained in another article [54], appears to follow a similar trend, with physically incorrect results for the larger basis sets. This article also presents an alternative population analysis that utilizes orbitals less sensitive to basis sets. Any systematic study of the basis set dependence of LPA was not found in the literature.
The conclusion drawn by the author is to avoid LPA, at the very least in analysis involving larger basis sets and/or large atoms, and it would appear as if the polarizing basis functions are those that distort the result. MPA would appear to be more stale in the choice of basis set, although care must be taken concerning
30 Results and discussion the issues mentioned in Section 4.6.3 and LPA may none the less be more suiting for some systems [55]. Dipole moment can only be utilized for smaller molecules of certain geometries, as electron redistributions may cancel.
Table 4.3. Electron transfer for the different molecules. The values are number of electron transfered in a bond, ZH− QH. Values obtained with B3LYP and basis set
given in the footnote.
Molecule Basis LPA MPA Dipole moment
FH A 0.0611 0.2100 – B -0.2407 0.3184 0.4132 H2O A 0.0472 0.1296 – B -0.1722 0.2179 0.3394 NH3 A 0.0427 0.0691 – B -0.0937 0.1531 0.2596 CH4 A 0.0350 0.0286 – B -0.0224 0.1016 0.0000 A = cc-pVDZ, B = cc-pVTZ
Further, considering the theories of electronegativity as in Section 4.6.1, there should be a linear connection between ∆χijand the electron transfer. This is given
in Figure 4.6, where the results from the dipole moments and MPA at cc-pVTZ is utilized. The electronegativities are given in Table 1.1, as taken from the Pauling scale. Note that methane cannot be a part of the dipole moment considerations, due to symmetries giving zero dipole moment.
Studying this Figure 4.6, it is seen that the difference in electronegativity could indeed be used as a measure of charge transfer, at the very least for this case. Note further that the definitions in [51] give that FH is an ionic bond, while the rest is of a sort called polar covalent bonds, which simply lies in the boundaries between covalent and ionic, being polar to some extent.
4.7
Inversion of ammonia
Quantum chemistry can be utilized not only to determine and study equilibrium geometries and properties, but also chemical reactions. In this example the inver-sion of ammonia will be studied, i.e., the transition of nitrogen from the equilibrium position above the plane of hydrogen, to the equilibrium position below the plane of hydrogen [56]. As the potential energy barrier would be expected to be mir-rored around the intermediate configuration, there would be expected to be a local
maximumthere.
The local maximum is known as the transition state, TS, and this is simply given as a unstable configuration in any chemical reaction. If the energy is high enough to pass the transition state, the reaction will occur, possibly resulting in the next minima. If the energy is not enought, the configuration will once again return to the initial equilibrium. This is disregarding possible tunneling effects.
4.7 Inversion of ammonia 31 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Difference in electronegativity, ∆χ = χ A − χB Electron transfer Dipole moment Mulliken Analysis
Figure 4.6. Charge transfer vs. electronegativity for molecules. Lines are linear ap-proximation. Values obtained using B3LYP/cc-pVTZ.
4.7.1
Potential barrier
The barrier of inversion is obtained by calculating the equilibrium geometry and the geometry of the transition state, and results are given in Table 4.4.
Energy calculation for nine intermediate states are then carried out by consid-ering the geometric structure of equilibrium and TS, constructing a process where the position of the nuclei are stepped up with a constant distance toward TS. The resulting PES for the full inversion along with explaining figures are given in Figure 4.7.
Table 4.4. Calculated equilibrium and transition state geometries for ammonia. Bond lengths given in ˚A and bond angles in degrees. Values obtained using B3LYP/cc-PVTZ.
Parameter Equilibrium Transition state
H–N–H 1.0141 0.9969
H–H 1.6249 1.7266
HNH 106.5 120.0
The barrier of inversion obtained is 8.46 mEhand should be compared with a
the-oretical value of 8.27 mEh[56]. The comparison to theory rather than experiment
32 Results and discussion Equil. TS Equil. 1 2 3 4 5 6 7 8 9 Geometry Energy, mE h
Figure 4.7. The process of inversion of ammonia and PES for this. Energy barrier obtained is 8.46 mEh, to be compared to experimental value of 8.27 mEh, [56]. Obtained
4.7 Inversion of ammonia 33 0.1 0.12 0.14 0.16 0.18 0.2 Geometry
Charge transfer, electrons
Equilibrium Transition0
1 2
Dipole moment, Debye
MPA Dipole moment
Figure 4.8.Dipole moment and Mulliken population analysis for the process of inversion of ammonia, with approximative linear fits. Values obtained using B3LYP/cc-pVTZ.
4.7.2
Electron distribution
Analogues to the methods in Section 4.6, the dipole moment and Mulliken pop-ulation analysis is evaluated for the transition state, the equilibrium and each of the intermediate geometries. Expected would be a constant electron population from MPA, as the electron transfer can be expected to be virtually constant for a molecule with so small a change in configuration (the bond distance is almost constant). With this assumed, the dipole moment obtained would be linearly de-clining, as the nitrogen approaches the plane spanned by hydrogen atoms with the same distance for each step. The result of this is given in Figure 4.8.
What is seen in the results is a electron transfer that increases with 0.026 e, as the molecule enters transition state. The dipole moment is seen to be declining rather linearly to zero, as transition state is of such a symmetry that no dipole moment is possible.
