Lectures on Modern Physics JiunnRen Roan 4 Oct

Lectures on Modern Physics Jiunn-Ren Roan 4 Oct. 2007

Atoms and Molecules The Hydrogen Atom The Schrödinger Equation for the Hydrogen Atom Quantization of Orbital Angular Momentum Quantum Number Notation Electron Probability Distributions The Zeeman Effect and Electron Spin The Zeeman Effect The Stern-Gerlach Experiment Electron Spin Selection Rules Spin-Orbit Coupling

Atoms and Molecules Many-Electron Atoms The Schrödinger Equation for the Helium Atom Independent-Electron Approximation Central-Field Approximation Hartree Approximation The Periodic Table Atomic Term Symbols Hund’s Rules for Ground-State Terms The Hydrogen Molecule The Schrödinger Equation for the Hydrogen Molecule The Valence-Bond Method The Molecular-Orbital Method Molecular Term Symbols

Atoms and Molecules Appendix Solving the Schrödinger Equation for the Hydrogen Atom References

The Hydrogen Atom The Schrödinger Equation for the Hydrogen Atom For the hydrogen atom the Schrödinger equation has a spherically symmetric potential energy: Hence, it is most convenient to work in spherical coordinates and write Solving this equation (see Appendix A), we find the energy is quantized: where is the reduced mass of the proton-electron system and is called the principal quantum number.

The Hydrogen Atom Quantization of Orbital Angular Momentum It can be shown that the finiteness of the wave function on the z-axis requires that the orbital angular momentum be quantized: where is called the orbital angular-momentum quantum number or orbital quantum number. Also, it can be shown that because of angular periodicity of the wave function, the z-component of the orbital angular momentum must be quantized as well: where is called the orbital magnetic quantum number or magnetic quantum number.

The Hydrogen Atom

The Hydrogen Atom Quantum Number Notation Because the quantized energy En is determined solely by the principal quantum number n, distinct quantum states with different quantum numbers may have the same energy. These degenerate states are often labeled with letters: l 0 1 2 3 4 5 ··· Label s p d f g h ··· The letters s, p, d, and f are the first letters of “sharp”, “principal”, “diffuse”, and “fundamental”, respectively, used in the early days of spectroscopy. Another widely used notation is n 1 2 3 4 ··· Shell K L M N ···

The Hydrogen Atom Electron Probability Distributions The radial probability distribution function P(r) is given by integrating over the angular variables: The electron is most likely to be found at the maximum of P(r), which for the states having the largest possible l for each n (such as 1 s, 2 p, 3 d, and 4 f states) occurs at n 2 a 0, where is the Bohr radius, the radius of the ground state in the Bohr model. In the atomic unit system, it is used as the length unit, called a bohr. For states without spherical symmetry, to reveal the angular dependence three-dimensional probability distribution function has to be used.

The Hydrogen Atom

The Hydrogen Atom

The Zeeman Effect and Electron Spin The Zeeman Effect (Zeeman, 1896) The quantization of angular momentum is confirmed experimentally by the splitting of degenerate states and the associated spectrum lines when the atoms are placed in a magnetic field—the Zeeman effect. m A charged particle orbiting an oppositely charged center generates a magnetic dipole moment m that is proportional to the angular momentum L: q The ratio g is called the gyromagnetic ratio. In the Bohr model, (m: magnetic moment; m: reduced mass). In a magnetic field B directed along the +z-axis, the potential energy associated with m is given by Thus, in a magnetic field the 2 l+1 degenerate states associated with a particular subshell are no longer degenerate but split into distinct energy levels according to where m. B is the Bohr magneton:

The Zeeman Effect and Electron Spin

The Zeeman Effect and Electron Spin

The Zeeman Effect and Electron Spin From D. A. Mc. Quarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

The Zeeman Effect and Electron Spin Only the so-called normal Zeeman effect can be explained by quantization of orbital angular momentum. In the anomalous Zeeman effect, the splitting does not follow the prediction. Heisenberg and Landé independently found that the anomalous Zeeman effect can be explained by introducing half-integer quantum numbers. While Heisenberg was strongly discouraged (for good reasons) by his mentor, Sommerfeld, and a close friend, Pauli, and did not publish it, Landé published it and got his name forever associated with the Zeeman effect.

The Zeeman Effect and Electron Spin The Stern-Gerlach Experiment (Stern and Gerlach, 1922) Passing a beam of silver atoms through an inhomogeneous magnetic field, Stern and Gerlach expected to see the 2 l+1 degenerate states split into an odd number, 2 l+1, of components. However, they were surprised to see that the beam split into only two components.

The Zeeman Effect and Electron Spin (Uhlenbeck and Goudsmit, 1925) Pauli postulated in 1925 that an electron can exist in two distinct states and introduced in a rather ad hoc manner a fourth quantum number to describe the two states. Although with this he could explain the Stern-Gerlach experiment, no interpretation was given to the fourth quantum number. Before long, two Dutch graduate students, Uhlenbeck and Goudsmit, proposed that the electron might behave like a spinning sphere of charge instead of a point particle and the spinning motion would give an additional spin angular momentum S and spin magnetic moment ms. According to this proposal, the spin angular momentum, like the quantized orbital angular momentum, is also quantized: where the spin quantum number s = ½, and what the Stern-Gerlach experiment measures is the z-component of the spin angular momentum: where the spin magnetic quantum number ms, the fourth quantum number introduced by Pauli, has two values +½ and -½, corresponding to the two orientations, up and down, respectively, of the spin angular momentum.

The Zeeman Effect and Electron Spin Like the relation between orbital angular momentum and magnetic moment, the spin magnetic moment is also proportional to the spin angular momentum: where the g-factor for electron gs is needed to obtain agreement with experimental observations. The g-factor for electron does not have classical analog. In 1928 Dirac developed a relativistic generalization of Schrödinger equation for electrons, which gave gs = 2, exactly. The observed value, however, differs from Dirac’s prediction by a very small amount: gs = 2. 00231930436170. A theory, quantum electrodynamics, developed from early 1930 s to 1950 s is able to give a value that agrees with the experimental value to 10 -13. Therefore, adding the contributions from the orbital motion and the intrinsic spin, which gives in a magnetic field In the Stern-Gerlach experiment, the silver atom was in an S-state (l = 0), so m must be 0, leading to two components corresponding to ms = ±½. Thus the Stern-Gerlach experiment directly confirmed the existence of electron spin.

The Zeeman Effect and Electron Spin From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

The Zeeman Effect and Electron Spin Selection Rules According to the interaction energy, inclusion of electron spin can further split the energy levels. For example, the 2 p state splits into five levels (taking gs = 2) instead of only three in the absence of electron spin; and the 1 s state now splits into two levels. At first sight, it appears that the spectrum corresponding to 2 p → 1 s would comprise all the possible transitions. However, this is not the case, because the emitted photon carries one unit (ħ) of angular momentum and therefore conservation of angular momentum requires the selection rules be held. The six allowed transitions for 2 p → 1 s (Dl = -1) are identical to those obtained without spin (normal Zeeman effect).

The Zeeman Effect and Electron Spin-Orbit Coupling When the magnetic field is not very strong, however, the allowed transitions for 2 p → 1 s exhibit additional splitting, resulting in anomalous Zeeman effect. This is because the magnetic moments associated with the orbital and spin angular momenta are coupled – the magnetic field created by the orbital motion interacts with the spin magnetic moment. If the external magnetic field is not strong enough to render the magnetic field created by the orbital motion completely negligible, then the interaction between L and S must be taken into account. The strength of this interaction is proportional to L·S. In a strong magnetic field, the coupling shifts all the levels with mms > 0 upward slightly and those with mms < 0 downward slightly, and also removes the remaining degeneracy in the 2 p state. This splits each of the outer lines into two closely spaced lines (anomalous Zeeman effect), which agree with experimental observations.

Many-Electron Atoms The Schrödinger Equation for the Helium Atom The Schrödinger equation for the helium atom is where the kinetic energy Ki depends only on the position of the i-th electron ri and the potential energy contains three terms, two electron-nucleus interactions and one electron-electron interaction -e -e r 2 r 1 Ze (Z = 2 for helium). The equation can be written as The wave function, of course, depends on both positions:

Many-Electron Atoms Independent-Electron Approximation The helium-atom Schrödinger equation has no known exact solution. Many approximations have been invented to tackle this difficult three-body problem. The simplest approximation is to neglect the electron-electron interaction: so that the two electrons do not interact and behave independently, leading to where the one-electron wave function , called an atomic orbital, satisfies the Schrödinger equation for a hydrogen-like atom: Substituting the wave function into the Schrödinger equation gives Thus, the total energy is the sum of the two one-electron energies: where

Many-Electron Atoms In this approximation the ground state of the helium atom is characterized by the wave function and energy which is larger than the true ground-state energy, -79. 0 e. V. Apparently, this is not a good approximation and there is much room for improvement. Applying this approximation to multi-electron atoms other than the helium atom seems rather straightforward. For example, the ground state of the three-electron lithium atom might be. However, this is not true, because the Pauli exclusion principle says that no two electrons in an atom can have the same set of quantum numbers (n, l, m, ms). Therefore, the spin wave function a and b, corresponding to ms = ½ and -½, respectively, must be included, forming the so-called spin orbitals: . Then the ground-state wave function for the helium atom should be where 1 sa and 1 sb are shorthand notations; the ground state of lithium has only two electrons in the 1 s orbital and the third electron must be in an n = 2 state.

Many-Electron Atoms The Pauli exclusion principle is a consequence of a fundamental theorem called the spin-statistics theorem: the wave functions of a system of indistinguishable half-integer-spin particles (fermions) are antisymmetric under interchange of any pair of particles, whereas the wave functions of a system of indistinguishable integer-spin particles (bosons) are symmetric under interchange of any pair of particles. Accordingly, the ground-state wave function of the helium atom must be the “anti-symmetrized form” of namely, It can be readily seen that as required by the spin-statistics theorem.

Many-Electron Atoms Central-Field Approximation A less trivial and more useful approximation is to assume that each of the atomic electrons moves independently of the others in a spherically symmetric potential energy Vc(r) that is produced by the nucleus and all the other electrons. For the helium atom, Because the overall effect of the electrons is to screen the nuclear Coulomb field, the effect becomes more appreciable at greater distances: Apparently, Vc(r) must be non-coulombic, in which the degeneracy between states of the same n and different l is removed. This is because the electrons with smaller l penetrate closer to the nucleus, seeing a more negative Vc(r). So, for a given n, the states of lowest l have the lowest energy. On the other hand, since Vc(r) is spherically symmetric, the degeneracy in m is not affected. In general, the same quantum numbers (n, l, m, ms) can be used to label states, but the energy now depends on both n and l. The restrictions on values of the quantum numbers are the same as before.

Many-Electron Atoms Hartree Approximation (Hartree, 1928) A method for obtaining a central field is given by Hartree: where is the charge density associated with the j-th electron. To solve the approximate Schrödinger equation, with one must know Vc(ri), but Vc(ri) in turn is determined by the wave function to be solved. Therefore, this equation can only be solved self-consistently. Fock correctly included spin wave functions into the Hartree approximation and obtained a better approximation called the Hartree-Fock approximation. Calculations using the Hartree-Fock approximation gives results that well agree with experimental observations.

Many-Electron Atoms Ar Electron-diffraction data Hartree-Fock calculation From I. N. Levine, Quantum Chemistry, 4 th ed. , Prentice Hall, Englewood Cliffs (1991).

Many-Electron Atoms The Periodic Table In non-coulombic fields, for a given n, the states of the lowest l have the lowest energy, but the degeneracy in m is kept intact. In some cases the intrashell splitting (same n, different l) is larger than the intershell splitting (different n), so that an “inversion” of level order occurs. Thus, 4 s < 3 d, 5 s < 4 d, and 6 s < 5 d < 4 f. The level ordering in this figure is common for neutral atoms. From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

Many-Electron Atoms

Many-Electron Atoms Atomic Term Symbols The ground-state configurations as given in the table do not completely specify the state of an atom with partly filled shells (also called open shells), because electrons with given n and l can be distributed among the different possible m and ms values. To completely specify the state, it is necessary to have additional information on m and ms, which is given by the so-called term symbols (“terms” in spectroscopic language means energy levels). For a group of k electrons, the total angular momenta and their z-components are given by where L, S, M, and MS are the corresponding quantum numbers. Addition of angular momenta in quantum mechanics is a complicated business. Fortunately, for the addition of two angular momenta L 1 and L 2, the rule is simple: This can be understood as lining up L 1 and L 2 parallel to obtain the greatest value of L and in the opposite direction to obtain the least value.

Many-Electron Atoms The possible values of the total orbital angular momentum quantum number L for the k-electron system, therefore, can be obtained by repeatedly applying the addition rule for two angular momenta. The result is If all the quantum numbers li are equal, Lmin is zero; if one of the li is larger than others, Lmin is given by orienting the other angular momenta to oppose it. The possible values of the total spin angular momentum quantum number S can be obtained similarly: If k is even, Smin = 0; if k is odd, Smin = ½. In addition to L and S, the total angular momentum is used to further distinguish states that have the same L and S values (there are totally (2 L+1)(2 S+1) such states). The possible total angular momentum quantum numbers are

Many-Electron Atoms The term symbol is written as and capital letters are used for L: l 0 1 2 3 4 5 6 7 8 ··· Label S P D F G H I K L ··· and the electron spin superscirpt 2 S+1 is read as follows: 2 S+1 1 2 3 4 5 6 ··· singlet doublet triplet quartet quintet sextet ··· corresponding to the fact that for L > S, the number of possible J levels is equal to 2 S+1 (called the multiplicity of the term). Thus, the term symbol for an atom with L = 3, S = 3/2, J = 5/2 is 4 F 5/2 and is read “quartet F five halves”. For closed shells and subshells, all orbitals with the same n and l are doubly occupied, so L = 0 and S = 0, giving 1 S 0. Thus, the contributions from completely filled shells or subshells are always 1 S 0 and can be ignored. For open shells, consider a carbon atom in the excited state 1 s 22 p 3 p as an example. The possible values are L = 2, 1, 0 and S = 1, 0, so the possible terms are 3 D 3, 2, 1 (L = 2, S = 1), 1 D 2 (L = 2, S = 0), 3 P 2, 1, 0 (L = 1, S = 1), 1 P 1 (L = 1, S = 0), 3 S (L = 0, S = 1), and 1 S (L = 0, S = 0). 1 0

Many-Electron Atoms For the ground-state configuration 1 s 22 p 2, l 1 = 1, l 2 = 1, s 1 = ½, s 2 = ½, Pauli exclusion principle limits the possible m and ms values, so that there are totally 15 possible combinations (remember that electrons are indistinguishable): From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970). Since the largest value of M is 2, and it occurs only with MS = 0, there must be a state with L = 2 and S = 0, i. e. , a 1 D 2 , which corresponds to (2 L+1)(2 S+1) = 5 combinations of M and MS. The remaining combinations has Mmax = 1, so L = 1 and M = 0, ± 1. Each of these M values occurs with a value of MS = 0, ± 1, so S = 1. Thus, the term is 3 P 2, 1, 0 and it corresponds to 9 combinations of M and MS. The remain only one combination: M = 0 and MS = 0, corresponding to L = 0 and S = 0, i. e. , 1 S 0.

Many-Electron Atoms For electrons in different subshells, called non-equivalent electrons, there is no restriction from the Pauli exclusion principle. Electrons in the same subshell (equivalent electron), on the other hand, must face the restriction imposed by the exclusion principle and, therefore, some terms derived for nonequivalent electrons are not possible. From I. N. Levine, Quantum Chemistry, 4 th ed. , Prentice Hall, Englewood Cliffs (1991).

Many-Electron Atoms Hund’s Rules for Ground-State Terms Knowing the terms, we can find which term characterizes the ground state by three empirical rules, called Hund’s rules, due to Hund: (a) The stability decreases with decreasing S, so the ground state has maximum spin multiplicity. (b) For a given value of S (or spin multiplicity), the state with maximum (c) L is most stable. (c) For given S and L, the minimum J value is most stable if there is a 1. single open subshell that is less than half-filled and the maximum J 2. is most stable if the subshell is more than half-filled. Thus, From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

Many-Electron Atoms From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

Many-Electron Atoms From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

The Hydrogen Molecule The Schrödinger Equation for the Hydrogen Molecule e 1 r 12 r 1 A e 2 r 2 B r 2 A r 1 B HA R HB The hydrogen molecule has two protons and two electrons, so the total energy contains the kinetic energy of relative motion of the nuclei with reduced mass mp/2 (mp = proton mass), kinetic energy of the two electrons, and Coulombic energy of the six particle-particle pairs: Because mp is much greater than the electron mass, in many cases the massive nuclei can be assumed to be stationary and the associated kinetic energy be neglected. This is called the Born-Oppenheimer approximation. With this approximation, the Schrödinger equation to be solved becomes For large R, all potential energies but and are small, so the r 12 e 1 equation becomes that of two non-interacting hydrogen atoms: whose ground-state solution can be easily found to be r 1 A e 2 r 2 B r 2 A r 1 B HA R HB

The Hydrogen Molecule The Valence-Bond Method (Heitler and London, 1927) A natural starting point for finding the solution of the Schrödinger equation is the large-R solution. Since the electrons are indistinguishable, there is no way to find out which is associated with which nucleus. Therefore, two equally valid solutions for large-R are e 1 r 12 r 1 A e 2 r 2 B r 2 A r 1 B HA R r 12 e 1 r 1 B r 1 A r 2 B HB HA R HB They are the ground-state solution for each of the two widely-separated, non-interacting hydrogen atoms: and therefore are the large-R solution to the Schrödinger equation for the hydrogen molecule: In the atomic unit system the magnitude of 2 E 1 s (27. 2 e. V) is used as the energy unit, called hartree.

The Hydrogen Molecule The valence-bond (or Heitler-London) method uses linear combination of the large-R solution as a trial function and requires it satisfy the Schrödinger equation This gives which can be expanded: where are called matrix elements. The coefficients c 1 and c 2 that minimize the energy E will give the best approximation to the true ground-state solution.

The Hydrogen Molecule The coefficients c 1 and c 2 that minimize the energy can be found from which gives A nontrivial solution exists if and only if The matrix elements Sij are computed as follows (remember the 1 s orbital is real):

The Hydrogen Molecule where r 12 e 1 r 1 A e 2 r 2 B r r 2 A 1 B integrates over all the space where the two 1 s orbitals, HA HB R one centered on nucleus A and the other on nucleus B, are simultaneously nonzero. In other words, S computes how much 1 s. A and 1 s. B overlap and thus is called the overlap integral. Computation of the matrix elements Hij utilizes the fact that and large-R solution of the Schrödinger equation with energy 2 E 1 s: and are the

The Hydrogen Molecule The integral Q defined in H 11 is e 1 r 12 r 1 A e 2 r 2 B r 1 A R e 2 e 1 r 2 B HB HA R r 12 r 1 A r 2 A r 1 B HA r 12 e 1 e 2 e 1 r 2 B r 1 A r 2 A r 1 B HB HA R r 12 e 2 r 2 B r 2 A r 1 B HB HA R It represents the classical Coulombic interaction of the charge clouds [1 s. A(1)]2 with nucleus B, of the charge cloud [1 s. B(2)]2 with nucleus A, of the charge cloud [1 s. A(1)]2 and [1 s. B(2)]2, and of the nuclei with one another, so Q is called the Coulomb integral. The integral J defined in H 12 is Since 1 s. A(1)1 s. B(1) is not an electron density in the ordinary sense, J cannot be interpreted as a classical electrostatic interaction of two charge clouds. HB

The Hydrogen Molecule The strictly quantum-mechanical quantity J can be written as This indicates that J arises as a result of exchanging electrons between the two nuclei, so J is called the exchange integral. With these matrix elements the condition for nontrivial coefficients becomes The minimized energy is or, relative to the energy of two isolated hydrogen atoms, Since DE+ has a minimum at a finite R, the two nuclei are in a bound state, forming a stable diatomic molecule. The corresponding bonding and antibonding wave functions are

The Hydrogen Molecule From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

The Hydrogen Molecule The electron density distributions are given by From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

The Hydrogen Molecule The Molecular-Orbital Method In the atomic-orbital approach to the electronic structure of many-electron atoms, one-electron wave functions satisfying the Schrödinger equation with an approximate potential energy such as are used to build up, under the constraint imposed by the exclusion principle, a many-electron atom’s configurations corresponding to the ground state, the first excited state, and so on. r. A HA For the electronic structure of many-electron molecules, the molecular-orbital (MO) method developed in the early 1930 s by Hund, Mulliken, and others is a generalization of the atomic-orbital method. To construct the electron configurations of the hydrogen molecule, the MO theory first considers the e corresponding one-electron molecule: H 2+, the simplest molecule. r. B The Schrödinger equation for H 2+ is R HB To find the ground-state configuration of H 2, the large-R ground-state wave functions for H 2+, 1 s. A (the 1 s orbital centered on nucleus A) and 1 s. B (the 1 s orbit centered on nucleus B), are an appropriate starting point.

The Hydrogen Molecule The trial function is the simplest example of the method of linear combination of atomic orbitals (LCAO). Like the valence-bond method, MO method minimizes the energy in and obtains where The corresponding bonding (s or sg, g: gerade is the German word for even) and antibonding (s* or su, u: ungerade = odd in German) orbitals are The energy curves are qualitatively similar to the valence-bond results, so are the electron density distributions.

The Hydrogen Molecule From D. A. Mc. Quarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

The Hydrogen Molecule s 1 s* From D. A. Mc. Quarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

The Hydrogen Molecule Finally, placing two electrons of opposite spins into the bonding orbital gives the ground state of the hydrogen molecule: This method of constructing molecular wave functions is known as the LCAO-MO method. A From A. L. Companion, Chemical Bonding, 2 nd ed. Mc. Graw-Hill, New York (1979). B

The Hydrogen Molecule The ground state obtained by valence-bond (VB) method is Therefore, up to normalization constants, the relation between them is where e 1 r 12 r 1 A e 2 e 1 r 2 B r 1 A R e 2 r 2 B r 2 A r 1 B HA r 12 HB HA R HB This suggests that the MO theory overemphasizes ionic feature, whereas VB theory ignores it.

The Hydrogen Molecule Because the s orbitals constructed so far are made out of 1 s orbitals, they are denoted by s 1 s (or sg 1 s) and s*1 s (or su 1 s). Additional MOs can be constructed from other kinds of AOs in a similar way. So 2 s. A ± 2 s. B gives s 2 s and s*2 s. Because the 2 s AO has a higher energy than the 1 s AO, the energy ordering is s 1 s < s*1 s < s 2 s < s*2 s. Constructed from 2 pz (or 2 p 0), the MO 2 pz, A ± 2 pz, B are symmetric about the inter-nuclear axis and so are s orbitals. They are designated by s 2 pz (or sg 2 pz) and s*2 pz (or su 2 pz). From I. N. Levine, Quantum Chemistry, 4 th ed. , Prentice Hall, Englewood Cliffs (1991).

The Hydrogen Molecule The MO constructed from 2 px, y (or 2 p± 1) has a nodal plane in both bonding and antibonding orbitals. AOs with one nodal plane are called p orbitals, so MOs with one nodal plane are called p (the Greek counterpart of p) orbitals. Unlike the s orbitals, here the bonding orbital changes sign upon inversion through the origin (i. e. , it is an odd function), whereas the antibonding orbital remains unchanged upon inversion, so for p orbitals the bonding orbital is ungerade and the antibonding orbital is gerade. The bonding and antibonding orbitals are denoted by p 2 px, y (or pu 2 px, y) and p*2 px, y (or pg 2 px, y). From I. N. Levine, Quantum Chemistry, 4 th ed. , Prentice Hall, Englewood Cliffs (1991).

The Hydrogen Molecule For most homonuclear diatomic molecules built of atoms of period 2 elements, an approximate ordering of the energy levels is, according to experiment, s 1 s < s*1 s < s 2 s < s*2 s < p 2 px = p 2 py < s 2 pz < p*2 px = p*2 py < s*2 pz so But, since the energy difference between the s 2 pz and p 2 px, y orbitals are very small and varies with the atomic number of the nuclei and the inter-nuclear separation, the other possible scheme is s 1 s < s*1 s < s 2 s < s*2 s < s 2 pz < p 2 px = p 2 py < p*2 px = p*2 py < s*2 pz Fortunately, many of the predictions of the two schemes are the same.

The Hydrogen Molecule Molecular Term Symbols The molecular term symbol is written where 2 S+1 as usual is the multiplicity and is the magnitude of the axial component (along the molecular axis) of the total orbital angular momentum. The following Greek letters, corresponding to the English letters (s, p, d, . . . ) for atomic orbitals, are used for L: L 0 1 2 3 4 ··· Label S P D F G ··· As an example, consider H 2: (sg 1 s)2. Both electrons have m = 0, so L = 0. Pauli exclusion principle requires that their spins must be opposite, giving S = 0. Thus, the term symbol for H 2 is 1 S (a singlet sigma state). It is easy to see that a closed subshell (each set of degenerate MO constitutes a molecular subshell) configuration has both S = 0 and L = 0 and gives rise to only a 1 S term. For a less trivial example, consider B 2: (sg 1 s)2(su 1 s)2(sg 2 s)2(su 2 s)2(pu 2 p)2. The only non-trivial contribution is from the two electrons in the open subshell pu 2 p.

The Hydrogen Molecule The two combinations with L = 2 have MS = 0, so S = 0 and the term is 1 D (singlet delta). The other four combinations all have L = 0 and MS = 0, ± 1, so S = 1 and the term is 3 S. Finally, the only remaining combination has L = 0 and MS = 0, so S = 0 and the term is 1 S. Hund’s rules apply to molecular electronic states as well, so the state with the largest spin multiplicity will be the ground state. Thus, the ground state of B 2 is a 3 S state.

The Hydrogen Molecule Superscripts + and –, and subscripts g and u can be used to indicate additional symmetric properties of the term. If the wave function changes sign upon reflection in a plane through the nuclei, a superscript – is supplemented; otherwise, + is used. Because for states with L ≠ 0, such a reflection always changes the sign of the axial component of the total orbital angular momentum, superscripts ± are used only for S states. Subscripts g and u are added to show the parity (symmetry under inversion through the origin) of the term. Terms arising from an electron configuration that has an odd number of electrons in MOs of odd parity are odd (u); all other terms are even (g). From I. N. Levine, Quantum Chemistry, 4 th ed. , Prentice Hall, Englewood Cliffs (1991).

Appendix Solving the Schrödinger Equation for the Hydrogen Atom To solve the equation with spherically symmetric potential energy, in spherical coordinates , first write Applying a method called separation of variables, we assume and obtain

Appendix The Schrödinger equation then becomes which can be written Collecting terms of different dependence, we obtain The only way for terms depending on different independent variables to be equal all the time is each term is a constant:

Appendix The angular part can be further separated: so (Nf is a normalization factor) and Note that periodicity in the azimuthal angle gives Let then the equation for the polar angle becomes

Appendix Power series are often used to solve differential equations. It will be very helpful to make the resultant recursion relation as simple as possible, i. e. , involving as few terms as possible. To achieve this goal, we substitute into the equation. With a little algebra, it is easy to find that if then This form will make the power series couple only two instead of three or more terms: the coefficient of the term is It can be shown that only if the series terminates at certain power can the solution be finite at w = ± 1. This requirement gives Therefore, we conclude that in which l is an integer.

Appendix The differential equation now becomes where Because differentiating the factor 1 -w 2 more than twice results in zero and if , then the differential equation becomes This can be simplified as which is known as Legendre’s differential equation. Its solution is the Legendre polynomials Pl(w). Putting things together, we obtain the solution (up to a normalization factor Nq) where is called an associated Legendre function.

Appendix The radial part now takes the form For large r, the bound-state (E < 0) solution satisfies so On the other hand, for small r, the differential equation requires that Therefore, we can try and substitute it into the differential equation to obtain

Appendix Again, the power series method leads us to conclude that for the series to terminate at somewhere and result in a finite F(r) for large r, we must have where n is an integer. In other words, the energy is quantized: where as before. The differential equation to be solved thus becomes which can be written

Appendix Solutions to the equation are called Laguerre polynomials Ln+l: so solutions to the equation are simply where is called the associated Laguerre polynomial. Finally, returning to the original unknown function R(r), we get where Nr is a normalization factor.

Appendix The wave function therefore can be written It can be shown that the normalization condition gives

References 1. H. D. Young and R. A. Freedman, Sears and Zemansky’s University Physics (Pearson, 2008) 12 th ed. 2. M. Karplus and R. N. Porter, Atoms and Molecules (Benjamin, 1970). 3. D. A. Mc. Quarrie, Quantum Chemistry (Oxford University Press, 1983). 4. L. I. Schiff, Quantum Mechanics (Mc. Graw-Hill, 1971). 5. I. N. Levine, Quantum Chemistry (Prentice Hall, 1991) 4 th ed.