Instructor background for the discussion points of Section 2

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1 Supplementary Information for: Orbitals Some fiction and some facts Jochen Autschbach Department of Chemistry State University of New York at Buffalo Buffalo, NY , USA Instructor background for the discussion points of Section 2 The appearance of orbitals in quantum theory is conveniently put into context by discussing the hierarchy of approximations that allow chemists to predict electronic structure from first principles. Here and elsewhere in this article we consider a many-electron system. Chemical phenomena are accurately described by the (relativistic (1)) time-dependent many-body Schrödinger equation OH D i.@ =@t/ for the coupled motion of electrons and nuclei, which one can write in a general form in terms of a set of coordinates for N electrons, r i, i = 1 to N, and M atomic nuclei R A with charges Z A e and A = 1 to M. In the following, r and R refer collectively to the set of electron and nuclear coordinates, respectively. A series of approximations and conceptual steps then unfolds. Electron orbitals are appear first under point (iii) and orbital energies appear first under point (viii). (i) Space-time variable separation: In the absence of time-dependent external fields (such as lasers in electronic and vibrational spectroscopy or radio waves in NMR experiments), the Hamiltonian OH is time-independent. Separation of variables allows then to write.ri RI t/ D.rI R/ exp. iet = / where satisfies the time-independent Schrödinger equation OH D E. The problem is then reduced to finding and the total energy E for stationary states. (ii) Electrons-nuclei separation: Typically, at this stage the Born-Oppenheimer (BO) approximation is invoked by which the coupled motions of electrons and nuclei are separated, based on the argument that often (not always) the timescales / energy level spacings for the electronic and nuclear motions differ by orders of magnitude. In its most drastic clamped nuclei version, the nuclear motion is completely ignored and the focus is on the electronic problem (el) only: OH.r/.r/ D.r/E. The nuclear positions enter the electronic problem in the form of a set of fixed parameters from which the potential energy term in the Hamiltonian is determined. The superscript el is dropped from here on and it is implicitly understood that the discussion is about the electronic Schrödinger equation. (iii) Electron-electron separation (orbitals): Separation of electronic coordinates means that if OH D P O i h i.r i / then one may choose D Q i i.r i /, which we call the Hartree wavefunction, or a suitable linear combination of such products, and it would be the exact wavefunction. The i are normalized functions that depend only on a single set of electronic coordinates r i D.x i ; y i ; z i /. They are called orbitals. The product wavefunction implies that the probability of finding an electron at some position is independent of where the other electrons are (uncorrelated probabilities). The correcthamiltonian for a system with two or more electrons contains electron repulsion terms of the form e 2 =.4" 0 /rij 1 (Coulomb s law) for each unique pair of electrons, 1

2 where r 1 ij D Œ.x i x j / 2 C.y i y j / 2 C.z i z j / 2 1=2. These inverse distances of two electrons are not separable into sums where each term depends on only either r i or r j. Therefore, OH P i O h i.r i / and the exact cannot be a product of orbitals or a simple linear combination of such products. In reality, the electron probabilities are correlated, which is of significant relevance in chemistry. However, one might use Q i i.r i / as a first approximation, while keeping the Hamiltonian in its exact form, and figure out a way to determine the unknown functions i by some criterion. This is the essence of molecular orbital theory. (iv) Space-spin coordinate separation: Experimental evidence shows that electrons have an additional property other than charge, mass, momentum, angular momentum, and so on the electron spin. The spin of a single electron is a generalized angular momentum characterized by a quantum number of 1/2 and projections of D =2 on a chosen quantization axis. If one ignores relativistic effects, which is an approximation that is reasonable for the description of atoms in the upper third of the periodic table, spin and spatial degrees of freedom can be treated separately. The nonrelativistic Hamiltonian in the absence of fields does not contain spin-dependent terms, but spin is important for proper orbital counting in step (v) and therefore one cannot simply ignore it. In orbital based theories one then uses spin-orbitals ' i. i / with generalized spatial-spin electron coordinates i D fr i ; i g. If the Hamiltonian does not afford spin dependent terms it is admissible to write ' i. i / D i.r i /. or ˇ/, where i.r i / is a spatial orbital which gets multiplied by an up or down spin-projection eigenfunction. It is important to realize that the arrows ", # usually drawn for and ˇ in orbital diagrams are symbolic only, indicating the sign of the spin projection onto a quantization axis. The arrows do not indicate the actual orientation of the spin vectors. (v) Pauli principle: The concepts discussed under this point do not represent a further approximation, but they are important and conveniently introduced at this point: Because one cannot follow an individual quantum particle s trajectory (see also Note 1), an important consequence is that the electrons in an atom or molecule are indistinguishable. This means that the probability density of finding the electrons at certain positions,, must be invariant under a permutation of electron labels. To be able to match theory and experimental data, Pauli postulated that for electrons and other fermions (half-integer spin particles) the wavefunction changes sign whenever the labels of two electrons are switched (Pauli principle), leaving invariant. At the level of orbital theory, the Hartree wavefunction D Q i ' i.r i / violates the Pauli principle because electron i is associated with a specific orbital, ' i, which may therefore serve to identify this particular electron. The Pauli principle requires that all electrons are described by all orbitals simultaneously. It is therefore incorrect to say electron or, as in the exam question quoted above,... adding the energies of each electron... because this conceptually associates specific electrons with specific orbitals. It is of course fine to say orbital in the proper context. In more complex theories that go beyond simple orbital wavefunctions, one-electron amplitude of -type might be a suitable phrase instead. (vi) Pauli exclusion principle: Continuing from the previous point, the wavefunction must be constructed in such a way that the wavefunction changes sign whenever two electron labels are interchanged. This requirement is fulfilled when the orbitals are used in a Slater-determinant to construct an approximate many-electron from a sum of N Š D N Hartree products where electron labels are permuted and signs are distributed suitably to enforce the Pauli principle. A determinant is zero when two rows or two columns are identical. Thus, the orbitals in the 2

3 s Figure 1: Representation of two orthonormal orbitals as two orthogonal vectors of unit length. Rotation by an angle gives an alternative set of orthonormal orbitals without changing the wavefunction. Any other pair of distinct vectors in the same plane can also be used. The example shows isosurfaces ( 0:03 a.u., B3LYP/DZP//BP/DZP) for a set of C C bonding and orbitals of ethene. Also shown are linear combinations with D 45 ı representing two equivalent banana bonds 0 1 D b 1 and 0 2 D b 2. Note that the orbital phases are arbitrary. determinant must all be different, otherwise there is no wavefunction. Since there are only two spin functions by which to distinguish orbitals by spin, " and #, a spatial orbital i can appear at most twice in the determinant (Pauli exclusion principle). Either the orbital appears in the determinant once, with a spin-up or spin-down factor, or it appears twice, once with spin-up and once with spin-down. If the same i is used three times then the many-electron wavefunction is zero, i.e. there would be no electrons. One says instead: an orbital can be occupied by at most two electrons. An unoccupied orbital is any function of 3D space that is not used in constructing (i.e. any function that cannot be expressed as a linear combination of the occupied orbitals). An alternative way of expressing the Pauli principle in MO theory is to say that all the orbitals used to construct the Slater determinant wavefunction must differ either in their spatial or spin part (or both). In atoms where high symmetry allows one to label the spatial orbitals by three quantum numbers n; l; m it then follows that any two orbitals must differ either in at least one of their n; l; m quantum numbers or in their spin-projections. This is a common, but more specialized, way of stating the famous Pauli exclusion principle. (vii) Orthonormal orbitals: In the Hartree-Fock method, the orbitals are chosen such that the approximate determinant wavefunction generates an energy expectation value as low as possible. A variational criterion is also applied in Kohn-Sham DFT. The energy minimization with respect to the orbital functions then leads to effective one-electron equations for the (spin) orbitals of the form.oh C OV / i D P j j " ji where Oh represents the kinetic energy and the electron-nucleus attraction, OV is an effective orbital repulsion operator, and the " ji are arbitrary Lagrange multipliers. Where do the " ji come from? The energy expectation value of a Slater determinant has of order.n Š/ 2 terms, an astronomically large number for larger N, of which almost all vanish if the orbitals are mutually orthogonal. A determinant wavefunction remains unchanged if one takes linear combinations of the orbitals. Thus, without loss of generality one can always choose the orbitals to be conveniently orthonormal, R i j dv D ı ij, which generates the most compact energy expression. However, one must then take this choice into 3

4 account and keep the orbitals orthonormal during the energy minimization, otherwise the energy expression becomes invalid. This constraint is represented by the Lagrangian multipliers. (viii) Canonical orbitals and orbital energies: The wavefunction and expectation values for observable properties of the system do not change upon taking linear combinations of the orbitals one is free to choose, implicitly and from the beginning, a set of orbitals that diagonalizes the matrix of Langrangian multipliers. This de-couples the orbital equations to yield.oh C OV / i D i " i which now resembles a one-electron Schrödinger equation with the effective one-electron Hamiltonian Oh D.Oh C OV /. This operator is usually referred to as a (canonical) Fock operator. The Langrange multiplier " i is identified with an eigenvalue of the Fock operator, the so-called orbital energy, and i is one of its eigenfunctions. This provides a convenient setup to solve the orbital equations as a set of eigenvalue equations. Koopmans has shown that in Hartree-Fock theory the eigenvalues of Oh are approximations to physical observables. In Kohn-Sham density functional theory (DFT) (2) the eigenvalues can also be identified with observables, in particular " D IP (ionization potential). (ix) Non-canonical orbitals: Once the canonical orbitals are determined, any linear combination gives the same. Most commonly used are linear combinations that preserve the attractive property of the orbitals forming an orthonormal set. An example is shown in Figure 1 where a set of two orthogonal orbitals of the ethene molecule is represented by two orthogonal vectors. A linear combination yields a set of alternative orbitals, 1 0 ; 0 2 which, in the figure, is also orthonormal. Any two vectors (orbitals) spanning the same plane (Hilbert space) give the same wavefunction, whether orthogonal or not. The set of = and the set of banana bonds are equally representative of the C=C double bond. (x) Basis set expansions: There is one more road block: Because of the kinetic energy, the orbital equations are second-order differential equations in the three position variables x; y; z of the orbitals. In order to solve such equations analytically one needs to be able to separate these variables additively in the operator. For a molecule, Oh includes electron-nuclear attraction terms in the potential of the form Z A e 2 =.4" 0 /r 1 A, where r 1 A D Œ.x X A/ 2 C.y Y A / 2 C.z Z A / 2 1=2 and R A D.X A ; Y A ; Z A / is the position of nucleus A. The problem is similar to the problem with the interelectronic distances. Save for a few specialized situations, there is no general coordinate system for molecules in which the variables can be separated and therefore exact explicit analytic solutions cannot be sought. Possible ways forward are either to attempt numerical solutions of the orbital equations in three dimensions, or as usually done to expand the orbitals in a set of known functions of 3D space and determine the expansion coefficients. In the latter case, i.r/ D P m m.r/c mi. The set of functions f m g is called basis set, and the C mi are the orbital coefficients. Any smooth and integrable function can be calculated to any desired accuracy if sufficiently many appropriately chosen basis functions are used. Solving the orbital equations then turns into a calculation of the set of orbital coefficients. (xi) Atom centered basis sets: How to select the basis set? Chemists intuitively think of molecules as composed of atoms and therefore it is natural to consider orbitals in molecules as composed of atomic orbital (AO) functions. This would then imply that a MO can be constructed as a linear combination of AOs (LCAO). LCAO as described in most textbooks should conceptually be associated with minimal basis set calculations where the smallest possible number of AO functions is used. As an example, for H 2 using one 1s AO for each hydrogen constitutes a minimal basis for describing the ground state. Minimal basis calculations ignore the fact that 4

5 atoms in molecules are deformed (polarized) and that atoms may be more or less extended in a molecule compared to the free atom. A good quality basis set accounts for these facts by providing several sets of basis functions for each atomic shell, with different radial extensions, as well as higher angular momentum functions that serve to polarize the formally occupied atomic shells in the molecule. Instead of atom-centered basis sets one may adopt any other set of functions of 3D space to express the orbitals of a molecule. It turns out that the chemist s intuition is good in the sense that atom centered basis sets require relatively few functions in order to obtain decent results for molecules from quantum chemical calculations. Moreover, such a basis set leads to an intuitive, albeit approximate, picture of how bonds are formed in molecules. At the same time, the notion of LCAO sends the wrong message to students because it conjures an image of what would be technically viewed as a minimal basis, which is far from sufficient to describe molecular electronic structures. Another problem is that a suitable AO in a molecule may be quite different from an orbital calculated for a free atom. The H 2 molecule is a case in point where the optimal exponent of the two 1s e r basis functions used in a minimal basis LCAO is significantly larger than the exponent of D 1 for the 1s hydrogen atom wavefunction (6). The functions used in AO basis sets are not orbitals calculated to describe the wavefunctions of hydrogen and many-electron atoms. Notes The best available quantum field theories may consider particle fields along with radiation fields, where electrons as indistinguishable particles are theoretical constructs. We adopt here a first-quantization wavefunction-for-particles framework (if needed with relativistic corrections), as usual in chemistry, with the understanding that this theoretical framework provides sufficient accuracy to deal with the bulk of chemical phenomena and molecular spectroscopy. Nuclear motion comes into play as soon as vibrations or nuclear dynamics in chemical reactions are to be discussed. Doing so at a quantum mechanical level introduces similar problems as discussed here for the solution of the electronic equation. For normalized orbitals, R i idv D 1, which then also gives a normalized Hartree wavefunction. In relativistic quantum chemistry, the orbitals are spinors with four or two complex components. Correlated motion is not unique to quantum mechanics but has long been known in classical physics, e.g. for the correlated motion of the planets in the solar system. Dealing with the correlation problem numerically is possible but requires significant computational resources and algorithmic developments. Correlated electron wavefunction methods typically utilize an uncorrelated wavefunction as a starting point for the computations, which is why orbitals are also ubiquitous in such methods. The approach in Kohn-Sham DFT is conceptually different (2). Kohn and coworkers introduced orbitals in order to obtain a reasonable approximation for the kinetic energy such as to allow studies of molecular binding energies with respect to atomic fragments. It has been found that the orbitals from Kohn-Sham DFT calculations carry physical meaning in some, usually approximate, context (3 5), just as those form Hartree-Fock calculations do (albeit not necessarily the same meaning). The context must be clearly stated. In principle, one should write the set of coordinates as fr i ; s i g where r i is the position vector and s i is the spin vector of an electron, both in real space. Because of the properties of angular momenta, the three components of s i cannot be known all at the same time, and the length of s is fixed. It is therefore sufficient to provide the values of =2 of the spin projection onto the arbitrarily chosen z axis. For particles with integral spin (bosons), the wavefunction does not change sign. As a result, there is no Pauli exclusion principle for boson systems. The orbital index is different from the electron index. We assume here that the set of orbitals is always ordered such that the two indices have the same value, to keep the notation convenient. If one is pedantic, it is only admissible to say orbital for linear systems with C 1v or D 1h symmetry where 5

6 ; ; ı; ; : : : are proper labels for the symmetry species. We follow the usual convention of identifying orbitals by their antisymmetry with respect to some actual or approximate global or local plane of symmetry in general molecules that are not necessarily linear, and even only approximately planar in the vicinity of the -bond. The criterion is motivated by the variational principle which states that the exact energy is below any energy expectation value calculated with the exact Hamiltonian and an approximate wavefunction. The expression is Oh D 2 2m e r 2 e2 Z A 4" 0 PA ra where r A is an electron nucleus distance. The effective electron repulsion in orbital-based ab-initio methods has two contributions. One of them has a classical analog and represents the continuous orbital charge distributions used to describe the electronic system repelling each other. This repulsion is a continuous-function analog of the Coulomb repulsion formula given under point (iii) of Section. The other term is referred to as exchange. Exchange-Coulomb terms arise from the nonclassical quantum statistics of electrons related to the Pauli principle and also take into account that electrons are point charges and do not repel themselves. A self-repulsion arises if only the classical Coulomb repulsion of continuous charge densities is used in the calculations. This happens in DFT with common approximate functionals. In spin-unrestricted MO calculations the spin- orbitals are orthogonal to the spin-ˇ orbitals due to the orthogonality of the spin functions. The spatial orbitals are orthogonal within the and ˇ sets individually, but not necessarily between the two sets. This may cause spin contamination. The procedure is valid as long as.oh C OV / is invariant under orbital transformations. This is usually the case. It should be kept in mind that the Lagrange multipliers " i are in principle arbitrary. Once a set of orthonormal orbitals is found, one can construct Oh D P i j ii" i h i j with arbitrary " i such that the equation Oh i D i " i remains satisfied. Another point is that solving the equations is not entirely trivial because the OV term in the Fock operator depends on the orbitals. The equation serves therefore mainly as a test whether a given set of orbitals minimizes the energy. This leads to the self-consistent field procedure where an initial set of orbitals is guessed and subsequently improved until it satisfies the orbital equations. In Figure 1 the orbital is a localized orbital formed from a linear combination of delocalized canonical orbitals which collectively represent the set of C C and C H bonds. This turns the integro-differential equations for the orbitals into a generalized matrix eigenvalue equation which can be solved particularly efficiently with digital computers. In ab-initio calculations of correlated wavefunctions with standard atom-centered one-electron basis functions, high angular-momentum are very important in order to approximate the electron cusps at r ij! 0. More suitable for an accurate description of the electron cusps are two-electron basis functions that depend explicitly on the interelectronic distance (F 12 methods). For instance, computational codes for solids often employ functions that are essentially the same as the wavefunctions of the 3D particle in a box. In order to obtain reasonable molecular orbitals one needs very many of such functions in the basis set. Literature Cited 1. Reiher, M.; Wolf, A., Relativistic quantum chemistry. The fundamental theory of molecular science, Wiley-VCH, Weinheim, Parr, R. G.; Yang, W., Density functional theory of atoms and molecules, Oxford University Press, New York, Baerends, E. J.; Gritsenko, O. V., A quantum chemical view of density functional theory, J. Phys. Chem. A 1997, 101, Chong, D. P.; Gritsenko, O. V.; Baerends, E. J., Interpretation of the Kohn-Sham orbital energies as approximate vertical ionization potentials, J. Chem. Phys. 2002, 116,

7 5. Stowasser, R.; Hoffmann, R., What do the Kohn Sham orbitals and eigenvalues mean?, J. Am. Chem. Soc. 1999, 121, Ruedenberg, K., The physical Nature of the chemical bond, Rev. Mod. Phys. 1962, 34,