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

Lectures on Modern Physics

Jiunn-Ren Roan

4 Oct. 2007

Atoms and MoleculesThe Hydrogen Atom

The Schrödinger Equation for the Hydrogen AtomQuantization of Orbital Angular MomentumQuantum Number NotationElectron Probability Distributions

The Zeeman Effect and Electron SpinThe Zeeman EffectThe Stern-Gerlach ExperimentElectron SpinSelection RulesSpin-Orbit Coupling

Atoms and MoleculesMany-Electron Atoms

The Schrödinger Equation for the Helium AtomIndependent-Electron ApproximationCentral-Field ApproximationHartree ApproximationThe Periodic TableAtomic Term SymbolsHund’s Rules for Ground-State Terms

The Hydrogen MoleculeThe Schrödinger Equation for the Hydrogen MoleculeThe Valence-Bond MethodThe Molecular-Orbital MethodMolecular Term Symbols

Atoms and MoleculesAppendix

Solving the Schrödinger Equation for the HydrogenAtom

References

The Hydrogen AtomThe 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 sphericalcoordinates and write

Solving this equation (see Appendix A), we findthe energy is quantized:

where is the reduced mass of the proton-electronsystem and

is called the principal quantum number.

Quantization of Orbital Angular MomentumThe Hydrogen Atom

It can be shown that the finiteness of the wave function on the z-axis requiresthat the orbital angular momentum be quantized:

where

is called the orbital angular-momentum quantum number or orbital quantumnumber.

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 quantumnumber.

The Hydrogen Atom

Quantum Number NotationThe Hydrogen Atom

Because the quantized energy En is determined solely by the principal quantumnumber n, distinct quantum states with different quantum numbers may havethe same energy. These degenerate states are often labeled with letters:

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

l 0 1 2 3 4 5 · · ·

Label s p d f g h · · ·

n 1 2 3 4 · · ·

Shell K L M N · · ·

Electron Probability DistributionsThe Hydrogen Atom

The radial probability distribution function P(r) is given by integrating overthe angular variables:

The electron is most likely to be found at the maximum of P(r), which for thestates having the largest possible l for each n (such as 1s, 2p, 3d, and 4f states)occurs at n2a0, where

is the Bohr radius, the radius of the ground state in the Bohr model. In theatomic unit system, it is used as the length unit, called a bohr.

For states without spherical symmetry, to reveal the angular dependence thethree-dimensional probability distribution function has to be used.

The Hydrogen Atom

The Zeeman Effect and Electron Spin

The quantization of angular momentum is confirmed experimentally by thesplitting of degenerate states and the associated spectrum lines when the atomsare placed in a magnetic field—the Zeeman effect.

A charged particle orbiting an oppositely charged center generates a magneticdipole moment that is proportional to the angular momentum L:

The ratio is called the gyromagnetic ratio. In the Bohr model,(: magnetic moment; : reduced mass). In a magnetic field B directed alongthe +z-axis, the potential energy associated with is given by

Thus, in a magnetic field the 2l+1degenerate states associated with a particularsubshell are no longer degenerate but split into distinct energy levels accordingto

where B is the Bohr magneton:

q

The Zeeman Effect (Zeeman, 1896)

http://nobelprize.org/nobel_prizes/physics/laureates/1902/zeeman-bio.html


From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).


Only the so-called normal Zeeman effect can be explained by quantization oforbital angular momentum. In the anomalous Zeeman effect, the splitting doesnot follow the prediction.

Heisenberg and Landé independently found that the anomalous Zeeman effectcan be explained by introducing half-integer quantum numbers. While Heisenbergwas strongly discouraged (for good reasons) by his mentor, Sommerfeld, and aclose friend, Pauli, and did not publish it, Landé published it and got his nameforever associated with the Zeeman effect.

http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-bio.html

http://en.wikipedia.org/wiki/Alfred_Lande

http://en.wikipedia.org/wiki/Alfred_Lande

http://math.ucr.edu/home/baez/spin/node9.html

http://en.wikipedia.org/wiki/Arnold_Sommerfeld

http://nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-bio.html

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


http://nobelprize.org/nobel_prizes/physics/laureates/1943/index.html

http://en.wikipedia.org/wiki/Walther_Gerlach

Electron Spin (Uhlenbeck and Goudsmit, 1925)Pauli postulated in 1925 that an electron can exist in two distinct states andintroduced in a rather ad hoc manner a fourth quantum number to describe thetwo 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, proposedthat the electron might behave like a spinning sphere of charge instead of apoint particle and the spinning motion would give an additional spin angularmomentum S and spin magnetic moment s.

According to this proposal, the spin angular momentum, like the quantizedorbital angular momentum, is also quantized:

where the spin quantum number s = ½, and what the Stern-Gerlachexperiment measures is the z-component of the spin angular momentum:

where the spin magnetic quantum number ms, the fourth quantum numberintroduced by Pauli, has two values +½ and -½, corresponding to the twoorientations, up and down, respectively, of the spin angular momentum.


http://en.wikipedia.org/wiki/George_Eugene_Uhlenbeck

http://en.wikipedia.org/wiki/Samuel_Abraham_Goudsmit

Like the relation between orbital angular momentum and magnetic moment, thespin magnetic moment is also proportional to the spin angular momentum:

where the g-factor for electron gs is needed to obtain agreement with experimentalobservations. The g-factor for electron does not have classical analog. In 1928Dirac developed a relativistic generalization of Schrödinger equation for electrons,which gave gs = 2, exactly. The observed value, however, differs from Dirac’sprediction by a very small amount: gs = 2.00231930436170. A theory, quantumelectrodynamics, developed from early 1930s to 1950s is able to give a value thatagrees 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 mmust be 0, leading to two components corresponding to ms = ±½. Thus theStern-Gerlach experiment directly confirmed the existence of electron spin.


http://nobelprize.org/nobel_prizes/physics/laureates/1933/dirac-bio.html

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


Selection RulesAccording to the interaction energy,

inclusion of electron spin can further split the energy levels. For example, the2p state splits into five levels (taking gs = 2) instead of only three in the absenceof electron spin; and the 1s state now splits into two levels. At first sight, itappears that the spectrum corresponding to 2p → 1s would comprise all thepossible transitions. However, this is not the case, because the emitted photoncarries one unit (ħ) of angular momentum and therefore conservation of angularmomentum requires the selection rules

be held. The six allowed transitions for 2p → 1s (l = -1) are

identical to those obtained without spin (normal Zeeman effect).


Spin-Orbit CouplingWhen the magnetic field is not very strong, however, the allowed transitionsfor 2p → 1s exhibit additional splitting, resulting in anomalous Zeeman effect.

This is because the magnetic moments associated with the orbital and spinangular momenta are coupled – the magnetic field created by the orbital motioninteracts with the spin magnetic moment. If the external magnetic field is notstrong enough to render the magnetic field created by the orbital motioncompletely negligible, then the interaction between L and S must be taken intoaccount.

The strength of this interaction is proportional to L·S. In a strong magneticfield, the coupling shifts all the levels with mms > 0 upward slightly and thosewith mms < 0 downward slightly, and also removes the remaining degeneracyin the 2p state. This splits each of the outer lines into two closely spaced lines(anomalous Zeeman effect), which agree with experimental observations.


Many-Electron AtomsThe 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 thei-th electron ri and the potential energy contains three terms,two electron-nucleus interactions and one electron-electroninteraction

(Z = 2 for helium). The equation can be written as

The wave function, of course, depends on both positions:

Ze

-e

-e

r1

r2

Many-Electron Atoms

Independent-Electron ApproximationThe helium-atom Schrödinger equation has no known exact solution. Manyapproximations 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, satisfiesthe 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 bythe wave function

and energy

which is larger than the true ground-state energy, -79.0 eV. Apparently, this isnot a good approximation and there is much room for improvement.

Applying this approximation to multi-electron atoms other than the helium atomseems rather straightforward. For example, the ground state of the three-electronlithium atom might be . However, thisis not true, because the Pauli exclusion principle says that no two electrons inan atom can have the same set of quantum numbers (n, l, m, ms).

Therefore, the spin wave function and , 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 1s and 1s are shorthand notations; the ground state of lithium has onlytwo electrons in the 1s 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 calledthe spin-statistics theorem: the wave functions of a system of indistinguishablehalf-integer-spin particles (fermions) are antisymmetric under interchange of anypair of particles, whereas the wave functions of a system of indistinguishableinteger-spin particles (bosons) are symmetric under interchange of any pair ofparticles. Accordingly, the ground-state wave function of the helium atom mustbe the “anti-symmetrized form” of

namely,

It can be readily seen that

as required by the spin-statistics theorem.

Many-Electron Atoms

Central-Field ApproximationA less trivial and more useful approximation is to assume that each of theatomic electrons moves independently of the others in a spherically symmetricpotential 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 Coulombfield, the effect becomes more appreciable at greater distances:

Apparently, Vc(r) must be non-coulombic, in which the degeneracy betweenstates of the same n and different l is removed. This is because the electronswith 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 thequantum 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 tobe solved. Therefore, this equation can only be solved self-consistently.

Fock correctly included spin wave functions into the Hartree approximationand obtained a better approximation called the Hartree-Fock approximation.Calculations using the Hartree-Fock approximation gives results that wellagree with experimental observations.

http://en.wikipedia.org/wiki/Douglas_Hartree

http://en.wikipedia.org/wiki/Vladimir_Fock

Many-Electron Atoms

Ar

Electron-diffraction dataHartree-Fock calculation

From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).

Many-Electron Atoms

The Periodic TableIn non-coulombic fields,for a given n, the states ofthe lowest l have the lowestenergy, but the degeneracyin m is kept intact.

In some cases the intrashellsplitting (same n, different l)is larger than the intershellsplitting (different n), sothat an “inversion” of levelorder occurs. Thus, 4s < 3d,5s < 4d, and 6s < 5d < 4f.

The level ordering in thisfigure is common for neutralatoms.


Many-Electron Atoms

Many-Electron Atoms

Atomic Term SymbolsThe ground-state configurations as given in the table do not completely specifythe state of an atom with partly filled shells (also called open shells), becauseelectrons with given n and l can be distributed among the different possible mand ms values. To completely specify the state, it is necessary to have additionalinformation 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-componentsare given by

where L, S, M, and MS are the corresponding quantum numbers. Addition ofangular momenta in quantum mechanics is a complicated business. Fortunately,for the addition of two angular momenta L1 and L2, the rule is simple:

This can be understood as lining up L1 and L2 parallel to obtain the greatest valueof 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 Lfor the k-electron system, therefore, can be obtained by repeatedly applying theaddition 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 thanothers, Lmin is given by orienting the other angular momenta to oppose it.

The possible values of the total spin angular momentum quantum number S canbe 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 aretotally (2L+1)(2S+1) such states). The possible total angular momentum quantumnumbers are

Many-Electron Atoms

The term symbol is written as

and capital letters are used for L:

and the electron spin superscirpt 2S+1 is read as follows:

corresponding to the fact that for L > S, the number of possible J levels is equalto 2S+1 (called the multiplicity of the term). Thus, the term symbol for an atomwith L = 3, S = 3/2, J = 5/2 is 4F5/2 and is read “quartet F five halves”.

For closed shells and subshells, all orbitals with the same n and l are doublyoccupied, so L = 0 and S = 0, giving 1S0. Thus, the contributions from completelyfilled shells or subshells are always 1S0 and can be ignored.

For open shells, consider a carbon atom in the excited state 1s22s22p3p as anexample. The possible values are L = 2, 1, 0 and S = 1, 0, so the possible termsare 3D3,2,1 (L = 2, S = 1), 1D2 (L = 2, S = 0), 3P2,1,0 (L = 1, S = 1), 1P1 (L = 1, S = 0),3S1 (L = 0, S = 1), and 1S0 (L = 0, S = 0).

l 0 1 2 3 4 5 6 7 8 · · ·

Label S P D F G H I K L · · ·

2S+1 1 2 3 4 5 6 · · ·

singlet doublet triplet quartet quintet sextet · · ·

Many-Electron Atoms

For the ground-state configuration 1s22s22p2, l1 = 1, l2 = 1, s1 = ½, s2 = ½, Pauliexclusion principle limits the possible m and ms values, so that there are totally15 possible combinations (remember that electrons are indistinguishable):

Since the largest value of M is 2, and it occurs only with MS = 0, there must bea state with L = 2 and S = 0, i.e., a 1D2 , which corresponds to (2L+1)(2S+1) = 5combinations of M and MS.

The remaining combinations has Mmax = 1, so L = 1 and M = 0, ±1. Each of theseM values occurs with a value of MS = 0, ±1, so S = 1. Thus, the term is 3P2,1,0 andit corresponds to 9 combinations of M and MS.

The remain only one combination: M = 0 and MS = 0, corresponding to L = 0 andS = 0, i.e., 1S0.


Many-Electron Atoms


For electrons in different subshells, called non-equivalent electrons, there is norestriction from the Pauli exclusion principle. Electrons in the same subshell(equivalent electron), on the other hand, must face the restriction imposed bythe exclusion principle and, therefore, some terms derived for nonequivalentelectrons are not possible.

Many-Electron Atoms

Hund’s Rules for Ground-State TermsKnowing the terms, we can find which term characterizes the ground state bythree empirical rules, called Hund’s rules, due to Hund:

(a) The stability decreases with decreasing S, so the ground state hasmaximum spin multiplicity.

(b) For a given value of S (or spin multiplicity), the state with maximumL is most stable.

(c) For given S and L, the minimum J value is most stable if there is asingle open subshell that is less than half-filled and the maximum Jis most stable if the subshell is more than half-filled.

Thus,


http://en.wikipedia.org/wiki/Friedrich_Hund

Many-Electron Atoms


The Hydrogen MoleculeThe Schrödinger Equation for the Hydrogen Molecule

The hydrogen molecule has two protons and two electrons, so thetotal energy contains the kinetic energy of relative motion of thenuclei with reduced mass mp/2 (mp = proton mass), kinetic energy ofthe two electrons, and Coulombic energy of the six particle-particlepairs:

Because mp is much greater than the electron mass, in many cases the massivenuclei can be assumed to be stationary and the associated kinetic energy beneglected. This is called the Born-Oppenheimer approximation. With thisapproximation, the Schrödinger equation to be solved becomes

For large R, all potential energies but and are small, so theequation becomes that of two non-interacting hydrogen atoms:

whose ground-state solution can be easily found to be

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12

The Hydrogen Molecule

The Valence-Bond Method (Heitler and London, 1927)A natural starting point for finding the solution of the Schrödinger equation isthe 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

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 thehydrogen molecule:

In the atomic unit system the magnitude of 2E1s (27.2 eV) is used as the energyunit, called hartree.

HA HB

e2

R

e1

r2A

r1A r2B

r1B

r12

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12

http://en.wikipedia.org/wiki/Walter_Heitler

http://en.wikipedia.org/wiki/Fritz_London


The valence-bond (or Heitler-London) method uses linear combination of thelarge-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 c1 and c2 that minimize the energyE will give the best approximation to the true ground-state solution.


The coefficients c1 and c2 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 1s orbital is real):


where

integrates over all the space where the two 1s orbitals,one centered on nucleus A and the other on nucleus B,are simultaneously nonzero. In other words, S computes how much 1sA and 1sB

overlap and thus is called the overlap integral.

Computation of the matrix elements Hij utilizes the fact that and are thelarge-R solution of the Schrödinger equation with energy 2E1s:

and

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12


The integral Q defined in H11 is

It represents the classical Coulombic interaction of the charge clouds [1sA(1)]2

with nucleus B, of the charge cloud [1sB(2)]2 with nucleus A, of the charge cloud[1sA(1)]2 and [1sB(2)]2, and of the nuclei with one another, so Q is called theCoulomb integral.

The integral J defined in H12 is

Since 1sA(1)1sB(1) is not an electron density in the ordinary sense, J cannot beinterpreted as a classical electrostatic interaction of two charge clouds.

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12


The strictly quantum-mechanical quantity J can be written as

This indicates that J arises as a result of exchanging electrons between the twonuclei, 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 E+ has a minimum at a finite R, the two nuclei are in a bound state,forming a stable diatomic molecule. The corresponding bonding and antibondingwave functions are


The electron density distributions are given by



The Molecular-Orbital MethodIn the atomic-orbital approach to the electronic structure of many-electronatoms, one-electron wave functions satisfying the Schrödinger equation withan 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.

For the electronic structure of many-electron molecules, the molecular-orbital(MO) method developed in the early 1930s by Hund, Mulliken, and others isa generalization of the atomic-orbital method. To construct the electronconfigurations of the hydrogen molecule, the MO theory first considers the

corresponding one-electron molecule: H2+, the simplest molecule.

The Schrödinger equation for H2+ is

To find the ground-state configuration of H2, the large-R ground-state wavefunctions for H2

+, 1sA (the 1s orbital centered on nucleus A) and 1sB (the 1s orbit centered on nucleus B), are an appropriate starting point.

HA HB

e

R

rArB

http://en.wikipedia.org/wiki/Hund

http://nobelprize.org/nobel_prizes/chemistry/laureates/1966/mulliken-bio.html


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 ( or g, g: gerade is the German word for even)and antibonding (* or u, u: ungerade = odd in German) orbitals are

The energy curves are qualitatively similar to the valence-bond results, so arethe electron density distributions.



1s

1s*


Finally, placing two electrons of opposite spins into the bonding orbital gives theground state of the hydrogen molecule:

This method of constructing molecular wave functions is known as the LCAO-MOmethod.

A B

From A. L. Companion, Chemical Bonding, 2nd ed. McGraw-Hill, New York (1979).


The ground state obtained by valence-bond (VB) method is

Therefore, up to normalization constants, the relation between them is

where

This suggests that the MO theory overemphasizes ionic feature, whereas VBtheory ignores it.

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12

HA HB

e1

R

e2

r1A

r2A r1B

r2B

r12


Because the orbitals constructed so far are made out of 1s orbitals, they aredenoted by 1s (or g1s) and *1s (or u1s). Additional MOs can be constructedfrom other kinds of AOs in a similar way. So 2sA ± 2sB gives 2s and *2s.Because the 2s AO has a higher energy than the 1s AO, the energy ordering is1s < *1s < 2s < *2s.

Constructed from 2pz (or 2p0), the MO 2pz,A ± 2pz,B are symmetric about theinter-nuclear axis and so are orbitals. They are designated by 2pz (or g2pz)and *2pz (or u2pz).




The MO constructed from 2px,y (or 2p±1) has a nodal plane in both bonding andantibonding orbitals. AOs with one nodal plane are called p orbitals, so MOswith one nodal plane are called (the Greek counterpart of p) orbitals. Unlikethe orbitals, here the bonding orbital changes sign upon inversion through theorigin (i.e., it is an odd function), whereas the antibonding orbital remainsunchanged upon inversion, so for orbitals the bonding orbital is ungerade andthe antibonding orbital is gerade. The bonding and antibonding orbitals aredenoted by 2px,y (or u2px,y) and *2px,y (or g2px,y).


For most homonuclear diatomic molecules built of atoms of period 2 elements,an approximate ordering of the energy levels is, according to experiment, 1s < *1s < 2s < *2s < 2px = 2py < 2pz < *2px = *2py < *2pz

so

But, since the energy difference between the 2pz and 2px,y orbitals are verysmall and varies with the atomic number of the nuclei and the inter-nuclearseparation, the other possible scheme is 1s < *1s < 2s < *2s < 2pz < 2px = 2py < *2px = *2py < *2pz

Fortunately, many of the predictions of the two schemes are the same.

Molecular Term SymbolsThe molecular term symbol is written

where 2S+1 as usual is the multiplicity and

is the magnitude of the axial component (along the molecular axis) of the totalorbital angular momentum. The following Greek letters, corresponding to theEnglish letters (s, p, d, ...) for atomic orbitals, are used for :

As an example, consider H2: (g1s)2. Both electrons have m = 0, so = 0.Pauli exclusion principle requires that their spins must be opposite, giving S = 0.Thus, the term symbol for H2 is 1 (a singlet sigma state). It is easy to see thata closed subshell (each set of degenerate MO constitutes a molecular subshell)configuration has both S = 0 and = 0 and gives rise to only a 1 term.

For a less trivial example, consider B2: (g1s)2(u1s)2(g2s)2(u2s)2(u2p)2. Theonly non-trivial contribution is from the two electrons in the open subshell u2p.


0 1 2 3 4 · · ·

Label · · ·


The two combinations with = 2 have MS = 0, so S = 0 and the term is 1 (singletdelta).

The other four combinations all have = 0 and MS = 0, ±1, so S = 1 and the termis 3.

Finally, the only remaining combination has = 0 and MS = 0, so S = 0 and theterm is 1.

Hund’s rules apply to molecular electronic states as well, so the state with thelargest spin multiplicity will be the ground state. Thus, the ground state of B2

is a 3 state.


Superscripts + and –, and subscripts g and u can be used to indicate additionalsymmetric properties of the term. If the wave function changes sign uponreflection in a plane through the nuclei, a superscript – is supplemented; otherwise,+ is used. Because for states with ≠ 0, such a reflection always changes thesign of the axial component of the total orbital angular momentum, superscripts± are used only for states.

Subscripts g and u are added to show the parity (symmetry under inversion throughthe origin) of the term. Terms arising from an electron configuration that has anodd number of electrons in MOs of odd parity are odd (u); all other terms are even(g).


AppendixSolving 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 equalall the time is each term is a constant:

The angular part can be further separated:

so

(N 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 helpfulto make the resultant recursion relation as simple as possible, i.e., involving asfew 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 solutionbe 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-w2 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 Legendrepolynomials Pl(w).

Putting things together, we obtain the solution (up to a normalization factor N)

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 terminateat 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

Appendix

References 1. H. D. Young and R. A. Freedman, Sears and Zemansky’s University Physics

(Pearson, 2008) 12th ed. 2. M. Karplus and R. N. Porter, Atoms and Molecules (Benjamin, 1970). 3. D. A. McQuarrie, Quantum Chemistry (Oxford University Press, 1983). 4. L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1971). 5. I. N. Levine, Quantum Chemistry (Prentice Hall, 1991) 4th ed.

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

Documents

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