Department of Physics, Chemistry and Biology

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1 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 Linköpings universitet, SE Linköping, Sweden

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3 Bachelor s Thesis LITH-IFM-G-EX 09/2112 SE Chemical bond analysis in the ten-electron series Thomas Fransson Adviser: Examiner: Patrick Norman IFM Patrick Norman IFM Linköping, 4 June, 2009

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5 Avdelning, Institution Division, Department Computational Physics Department of Physics, Chemistry and Biology Linköpings universitet, SE Linköping, Sweden Datum Date Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LITH-IFM-G-EX 09/2112 SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version Titel Title Chemical bond analysis in the ten-electron series Författare 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 chemistry. 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 nature 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 Waalsinteraction resulting in a Ne 2 molecule is studied with a theoretical bond length of 3.23 Å and dissociation energy of µe h. The equilibrium geometries of FH, H 2O, NH 3 and CH 4 are studied and the strength and character of the bonds involved evaluated using bond order, dipole moment, Mulliken population analysis and Löwdin population analysis. The concept of electronegativity is studied in the context of electron transfer. Lastly, the barrier of inversion for NH 3 is studied, with an obtained barrier height of 8.46 me h and relatively constant electron transfer. Nyckelord Keywords computational chemistry, computational physics, quantum chemistry, bond analysis, electron population, electron transfer

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7 Abstract This thesis presents briefly the application of quantum mechanics on systems of chemical interest, i.e., the field of quantum chemistry and computational chemistry. 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 nature 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 Waalsinteraction resulting in a Ne 2 molecule is studied with a theoretical bond length of 3.23 Å and dissociation energy of µe h. The equilibrium geometries of FH, H 2 O, NH 3 and CH 4 are studied and the strength and character of the bonds involved evaluated using bond order, dipole moment, Mulliken population analysis and Löwdin population analysis. The concept of electronegativity is studied in the context of electron transfer. Lastly, the barrier of inversion for NH 3 is studied, with an obtained barrier height of 8.46 me h and relatively constant electron transfer. v

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9 Acknowledgements First of all, I woulds like to thank my supervisor Patrick Norman for the opportunity 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 Computational 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. vii

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11 Contents 1 Introduction Quantum chemistry Relevant elements and molecules Electronic structure theory Many-particle systems The Born Oppenheimer approximation Antisymmetry The Hartree Fock approximation Basis consideration Molecular orbitals and applications Construction of molecular orbitals Molecular orbitals Electron density Potential energy surface Electron correlated methods Post-Hartree Fock methods Density functional theory Hybrid methods Molecular properties Computational details Programs used Basis sets used Methods used Results and discussion Neon monomer Orbitals Electron density Orbital energies Neon dimer Basis set superposition error Hartree Fock Second-order Møller Plesset ix

12 x Contents Perturbation theory Results Molecular equilibrium geometries Orbital analysis Hybridization Natural atomic orbitals Bond order Bond character Electronegativity Dipole moment Mulliken population analysis Löwdin population analysis Results Inversion of ammonia Potential barrier Electron distribution Conclusions 35 Bibliography 37

13 Chapter 1 Introduction 1.1 Quantum chemistry The branch of physics or theoretical chemistry known as quantum chemistry attempts 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 treatment 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 heavier 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 interactions. 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 meter, respectively) and as molecular analysis works at a scale of about 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 reactions too unstable for experiment [9]. 1

14 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 Carbon C 1s 2 2s 2 2p Nitrogen N 1s 2 2s 2 2p Oxygen O 1s 2 2s 2 2p Fluorine F 1s 2 2s 2 2p Neon Ne 1s 2 2s 2 2p 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/m 3 and temperature in K. Formula Name Density Melting point a Boiling point a Ne Neon c Ne 2 Neon dimer FH d Hydrogen fluoride H 2 0 Water 998 b NH 3 Ammonia 0.77 c CH 4 Methane 0.72 c Values obtained from [2]. a at 10 5 Pa. b at 293 K and 10 5 Pa. c at 273 K and 10 5 Pa. d obtained from [13].

15 Chapter 2 Electronic structure theory The main problem in quantum chemistry is that it deals with larges system, composed 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 Schrödinger equation for a many-particle wave function Ĥ Ψ = E Ψ (2.1) where Ĥ 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 r is ρ( r) = Ψ( r) 2 d 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 approximations 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 Ĥ = N i= i M A=1 1 N 2 2M A A M i=1 A=1 3 Z A r ia + N i=1 j>i 1 r ij + M A=1 B>A Z A Z B R AB (2.3)

16 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 The Born Oppenheimer approximation A first approximation that is generally made and of central importance to quantum 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 configurations, 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, Ĥ = N i= Antisymmetry i N M i=1 A=1 Z A r ia + N i=1 j>i 1 r ij (2.4) 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 r is the spatial coordinates and ω is the spin function. A common procedure is to divide each orbital into a spatial- and a spindependent 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 orthonormal basis Ψ i ( r) Ψ j ( r) = δ ij (2.7) and the spin functions are assumed to follow α α = 1; β β = 1; α β = 0; β α = 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

17 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( x 1,..., x i,..., x j,..., x N ) = Ψ( x 1,..., x j,..., x i,..., x N ) (2.9) A mathematical function that follows this antisymmetric property is a determinants, 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( x 1, x 2,..., x N ) = 1 N! χ i ( x 1 ) χ j ( x 1 ) χ k ( x 1 ) χ i ( x 2 ) χ j ( x 2 ) χ k ( x 2 ) χ i ( x N ) χ j ( x N ) χ k ( x N ) (2.10) 2.2 The Hartree Fock approximation The simplest physically acceptable wave function is a single SD, i.e., Ψ = χ 1 χ 2 χ N (2.11) Using this and the full electronic Hamiltonian given by Equation 2.4, the variational principle [14] thus gives, E = Ψ Ĥ Ψ E 0 (2.12) where E 0 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, Ψ orbital i = ν c νi φ trial ν (2.13) so that the c are variational parameters to be weighted in such a way that a minimum is achieved in 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

18 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 description 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, Ψ MO i = ν c νi φ AO ν (2.14) where the set of c νi are 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 computationally 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 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.) N N N Ô 1 = ĥ(i) and Ô 2 = ˆV (i, j) (2.16) i=1 i=1 j=1 If the operators are spin-independent, as in non-relativistic considerations, the operations of such yields N Ψ Ô1 Ψ = 2 h kk (2.17) resulting in one-electron integrals, h kk, and N/2 Ψ Ô2 Ψ = 1 (2 Ψ i Ψ j ˆV Ψ i Ψ j Ψ i Ψ j ˆV Ψ j Ψ i = 1 2J ij K ij (2.18) 2 2 i,j k=1 N/2 i,j

19 2.4 Molecular orbitals and applications 7 where J ij is known as the Coulomb integral and K ij is known as the exchange integral. It is then convenient to express corresponding operators for these integrals χj ( x 2 ) 2 Ĵ j χ i ( x 1 ) = [ r 1 r 2 d x 2]χ i ( x 1 ) (2.19) χ j ( x 2 )χ i ( x 1 ) ˆK j χ i ( x 1 ) = [ d x 2 ]χ j ( x 1 ) (2.20) r 1 r 2 so that J ij = χ i Ĵj χ i (2.21) K ij = χ i ˆK j χ i (2.22) (2.23) Using operators from 2.16, 2.19 and 2.20, the Fock operator can be formed, as N ˆf = ĥ + (Ĵj ˆK j ) (2.24) j 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, ˆf χ i = N ǫ ji χ j (2.25) j 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 ˆf χ i = ǫ i χ i (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 Theorem. The opposite, electron affinity, is true when considering virtual (unoccupied) orbitals, E N 1 E N = ǫ a and E N E N+1 = ǫ v (2.27) where ǫ a and ǫ v are the orbital eigenvalues for the occupied and virtual orbital, respectively.

20 8 Electronic structure theory 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 Ψ i = ν c νi φ ν Ψ = φ T c (2.28) where c is a matrix formed by the MO-coefficients. While the molecular orbitals are orthonormal (as by Equation 2.7), orthonormality or even orthogonality is typically not achieved for the atomic orbitals used, but the following is true Ψ i Ψ j = α,β c αic βj φ i φ j = δ ij (2.29) where the inner product between two arbitrary atomic orbitals is called the overlap matrix, S, as S αβ = φ α φ β (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, S αβ = φ α φ β = φ β φ α = S βα (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 E but csc = E (2.32) Further, symmetrically orthogonalized atomic orbitals may be constructed from the atomic orbitals simply by considering Electron density φ = S 1/2 φ (2.33) 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 γ k ( r 1,.., r k, r 1,.., r k) = Ψ( x 1,.., x N )Ψ ( x 1,.., x N)dω 1..dω k d r k+1..d r N (2.34)

21 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( r 1, r 1) = N Ψ( x 1,.., x N )Ψ ( x 1,.., x N)dω 1 d x 2..d x N (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 r 1. The integral of this density function thus equals the total number of electrons. ρ 1 ( r 1 ) = γ( r 1, r 1 ) = N Ψ( x 1,.., x N )Ψ ( x 1,.., x N )dω 1 d r 2..d r N (2.36) N = ρ 1 ( r 1 )d r 1 (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 occ occ ρ HF ( r) = γ HF ( r, r) = 2 Ψ i ( r)ψ i ( r) = 2 c αi φ α ( r) i i α β = occ [2 c αi c βi]φ α ( r)φ β ( r) = Dαβ AO φ α ( r)φ β ( r) αβ i αβ c βiφ β( r) (2.38) where D AO 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, occ N = 2 Ψ i ( r) 2 d r = D αβ S βα (2.39) α,β i where, for simplicity, we have denoted D AO = D. It may also be constructed a density matrix in MO-basis, this should follow (remember that this is a closed-shell molecule) D MO = { 2δij j occ 0 otherwise (2.40) Following Equation 2.28, the two density matrices can be connected by D = cd MO c (2.41) and, if considering Equation 2.32 D MO = c SDSc (2.42)

22 10 Electronic structure theory Potential energy surface Having obtained the molecular orbitals for an arrangement of the nuclei, the energy associated with this can be determined. By computing the energy 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] 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 E corr = E 0 E HF (2.43) where E HF is the Hartree-Fock limit energy. Th energy of the better method is denoted E 0, 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 Determinants 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 combination 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].

23 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] 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 functionals, 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] 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,...) = E 0 + E F F + E B +... (2.44) B where E 0 is the energy of the isolated system and F, B correspond to applied electronic field and magnetic field, respectively. Other perturbing fields are possible but will not be considered. E 0 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 atomic units. Molecular properties are defined as the partial derivatives in 2.44 given at zero field strength. Further, the properties will be static if the perturbing fields are

24 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 m 0 i = E B i permanent magnetic dipole moment (2.45) F,B, =0 µ 0 i = E F i permanent electric dipole moment (2.46) F,B, =0 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 ε(a,λ) (2.47) λ 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 = Ψ Ĥ A Which is known as the Hellmann Feynman theorem [19]. Ψ (2.49)

25 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 calculations, 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 hybridization 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 corresponding 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 ccpvtz, 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 13

26 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.

27 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 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 described 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) 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. 15

28 16 Results and discussion 1s 2s 2p y 2p x 2p z Figure 4.1. Occupied ground-state orbitals of Neon. The surface represent the isoelectric 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 radius for 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 Orbital energies Considering Koopmans theorem, Equation 2.27, the ionization energies of canonical 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

29 4.1 Neon monomer s 2s (lower curve) 2p Full density 6 Electron density, ρ Radial distance, (Ã) 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.

30 18 Results and discussion 0 1 2p (6) 2p /2 (2) 2p /2 (4) 2 2s (2) 2s (2) 3 Energy, E h s (2) 1s (2) 34 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.

31 4.2 Neon dimer 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 interactions 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] 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 different compounds energies in their own basis sets with the energy of the different 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].

32 20 Results and discussion 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 interaction of an electron with an effective charge cloud, thus preventing any correlation. The calculation is carried out as an illustration 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 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 compared 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], E NeNe vdw = 3 πr 6 0 [ᾱ Ne (iω)] 2 dω = C 6 R 6 (4.3) The value of C 6 was found using HF/taug-cc-pVTZ, with a numerical value of 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 MP Results Using these different methods, the potential energy surface was calculated according to Section 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 a.u. ( 2 4 Å). 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 Å). The BSSE for MP2 is considerable, with a value at equilibrium of about µe h. Finally, MP2 gives the best results, with a potential minima of 81.8 µe h at 6.10 a.u. (3.23 Å) separation. This should be compared to the experimental values

33 4.2 Neon dimer Energy, me h MP2 HF PT BSSE Interatomic distance, a.u. 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.

34 22 Results and discussion of about 127 µe h and 5.84 a.u. (3.09 Å) [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 Ne 2 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 Å and bond angles in degrees. Molecule Parameter Theory Experiment a Ne 2 Ne Ne FH b H F H H 2 O c H O H HOH c NH 3 H N H HNH c CH 4 H C H HCH d a obtained from MP2/cc-pVTZ according to Section 4.2, experimental value refer to [32] ([37] have 3.1 Å). b obtained from [37]. c obtained from [38]. d value 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 suitable 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.

35 4.4 Orbital analysis 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, Ω = φ iφ i Ω φ iφ i (4.4) N basis i=1 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 calculation 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 Ω Boys = φ iφ i ( r 1 r 2 ) φ iφ i (4.5) N basis i=1 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 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 sp 3 -orbitals, i.e. hybrids formed by a combination of 1s from hydrogen, and a sp 3 from the carbon. sp 3 is a hybrid orbital formed by the combination of 2s and all 2p:s Natural atomic orbitals Consider the density matrix D (in AO-basis). This matrix is given from Equation 2.4.3, as occ occ D = 2 c i c i = 2 (c i c i ) = D (4.6) i=1 and it is thus Hermitian. Observe again that closed-shell molecules are considered. 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. i=1

36 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