Many-electron atomsIn constructing the hamiltonian operator for a many electron atom, we shall assume a fixed nucleus and ignore the minor error introduced by using electron mass rather than reduced mass. There will be a kinetic energy operator for each electron and potential terms for the various electrostatic attractions and repulsions in the system. Assuming n electrons and an atomic number of Z, the hamiltonian operator is (in atomic units):

1,2,3,spatial coordinates of each of the n electrons. 1x1,y1,z1, or r1, 1, 1,(),, 1/r12, 1/r21,, 1/r22 case, He atom , hamiltonian :

is called one-electron operator, and 1/r12 is called two-electron operator.,,,

Where n is the principal quantum number for atomic orbital i , and Z is the atomic nuclear charge in atomic units. i (1) where 1 is the position coordinate of electron 1, and the atomic orbital i is used for any one-electron fN for describing the electronic distribution about an atom.Products of the atomic orbitals is are eigenfNs of Happ, and the eigenvalue E is equal to the sum of the atomic orbital energies is

Simple products and electron exchange symmetryFor the configuration 1s12s1, the wavefN is :12,, so that it yields an average value for r1 and r2 that is independent of our choice of electron labels. This means that the electron density itself must be independent of our electron labeling scheme.

,, 1s(1)2s(2) and 1s(2)2s(1) ,,12 symmetric to the exchange of labels, antisymmetric to the exchange of labels.

Electron spin and the exclusion principleStern and Gerlach observed two bands of Ag atom in their expt.

spin,and ,normalized spin fNs,Pauli principle: wavefNs must be antisymmetric with respect to simultaneous interchange of space and spin coordinates of electrons, called spin-orbitals of electrons.

Slater determinantsSlater suggested that there is a simple way to write wavefNs guaranteeing that they will be antisymmetric for interchange of electronic space and spin coordinates, for example 1s12s1:

For three electrons wavefNs, 1s22s1,,, antisymmetric to the exchange of any two electrons indices spin-orbitals, indices,1,2,3

Singlet and triplet states orbital paired () total spin S0 2S+1=1 2S+1=1 spin-multiplicity 1 singlet2doublet3triplet4quartetorbitalssinglet triplet ,excited state of He, 1s12s1,Pauli principle wavefNs:

, triplet state

equivalent to (1) equivalent to (2) and (4)The lesson to be learned from this is that a single Slater determinant does not always display all of the symmetry possessed by the correct wavefN.

Paired spin, s=0,Ms=0, xyThis is calledSinglet.

Two electrons with parallel spins,have a nonzero total spin angularmomentum. the angle between the vectors is the same in all three cases:the resultant of the two vectors have the same length in each case, but points in different directions. paired spin two paired spin are preciselyantiparallel, however, two parallelspins are not strictly parallel.This is called triplet

Next we will investigate the energies of the states as they are described by these wavefNsWe have known that they are eigenfNs of Happ, but not eigenfNs of real hamiltonian, therefore, we calculate the average values of the energy for the singlet and triplet state wavefNs:

We notice that the hamiltonian operator has no interaction term on spin part, this means that the average energy will be entirely determined by the space parts. Therefore, the triplet state will have the same energy, but that of the singlet state may have a different energy. Which of these two state energies should be higher?,,,:

:

The orthogonality of the 1s and 2s orbitals caused the terms preceded by to vanish. Furthermore, integrals that differ only in the variable label ( such as those in the 2nd and 3rd terms )are equal.

:

so that this expansion becomes,expansion over (-2/r1, -2/r2),

,1/r12, occurs in four two-electron integrals:

Thus, the average energy value is:The first two terms gives the average energy of He+ in its 1s state, and the second pair gives the energy of He+ in 2s state, thus the final becomes,

Where J and K represent the last two integrals. The integral J denotes electrons 1 and 2 as being in charge clouds described by 1s*1s and 2s*2s, respectively. The operator 1/r12 gives the electrostatic repulsion energy between these two charge clouds.,J, coulomb integral.

K is called an exchange integral because the two product fNs in the integrand differ by an exchange of electrons.

Kthe interaction between an electron distribution described by 1s*2s, and another electron in the same distribution. (,)r1 and r2 are both smaller or Both larger, then the fN 1s(1)2s(1)1s(2)2s(2)will be positive. But when one r value is smaller than R and the other is largerthan R, on opposite sides of the nodalsurface, then 1s(1)2s(1)1s(2)2s(2)is negative. These positive or negative contributionsto K are weighted by the fN of 1/r12,K,,,

Since the integral K is positive, we can see that from the derived equation that the triply degenerate energy level lies below the singly nondegenerate one, the separation between them being 2K.

What is the meaning of Fermi holeIn triplet state the space part of the wavefN:What would happen if these two electrons are collide ? Which means that the coordinate of 1 electron is equal to 2 electron, that is, 1s(1)=1s(2), and 2s(1)=2s(2), so that, the above equation should be vanished. That means, this situation should never happen. This situation is called Fermi hole, and it is built into any wavefN that is properly antisymmeterized.

singlet state (symmetric space fN) coordinate , 1s(1)=1s(2), wavefN vanish, (spin ), coulomb hole. However, wavefN vanish. Why? It is due to our independent-electron approximations (that is, the electrons were attracted by the nucleus but somehow did not repel each other).

Warning: Usable approximations to eigenfNs are very useful in understanding, predicting, and calculating observable phenomena. But one must always be aware of the possibility of significant differences existing between the real system and the mathematical model for that system. triplet state wavefN Fermi hole ,,, basis fN,basis fN,r12,(Table 5-1, wavefN,, 1/r12 singlet and triplet stateswavefN),:

suppose we take ordinary independent-electron wavefN as our initial approximation for the helium atom: They are correct only if electrons 1 and 2 do not see each other via a repulsive interaction. However, this is not the true case. How are we going to correct it?

The Self-Consistent Field, Slater-Type Orbitals, and the Aufbau Principlewe can approximate this repulsion by saying that electron 1 sees electron 2 as a smeared out, time-averaged charge cloud rather than the rapidly moving point charge which is actually present. The initial description for this charge cloud is just the absolute square of the initial atomic orbital occupied by electron 2: [1s(2)] 2.

Our approximation now has electron 1 moving in the field of a positive nucleus embedded in a spherical cloud of negative charge by electron 2. Thus, for electron 1, the positive charge is shielded or screened by electron 2. Hence electron 1 should occupy an orbital that is less contracted about the nucleus. Let us write this new orbital in the form:Where is related to the screened nuclear charge seen by electron 1. Next we turn to electron 2, which we now take to be moving in the field of the nucleus shielded by the charge cloud due to electron 1, now in its expanded orbital. Just as before, we find a new orbital of form (1) for electron 2. Now, however, will be different because the shielding of the nucleus by electron 1 is different from what was in our previous step.

We now have a new distribution for electron 2, but this means that we must recalculated the orbital for electron 1 since this orbital was appropriate for the screening due to electron 2 in its old orbital. After revising the orbital for electron 1, we must revise the orbital for electron 2. This procedure is continued back and forth between electrons 1 and 2 until the value of converges to an unchanging value (under the constraint that electrons 1 and 2 ultimately occupy orbitals having the same value of ). Then the orbital for each electron is consistent with the potential due to the nucleus and the charge cloud for the other electron: the electrons move in a self-consistent field (SCF).

The result of such a calculation is a wavefN in much closer accord with the actual charge density distributions. However, because each electron senses only the time-averaged charge cloud of the other in this approximation, it is still an independent-electron treatment.

The hallmark() of independent electron treatment is a wavefN containing only a product of one-electron fNs. There are no fNs of, say, r12, which would make wavefN depend on the instantaneous distance between electrons 1 and 2.Atomic orbitals that are eigenfNs for the one-electron hydrogenlike ion are called hydrogenlike orbitals. Since these orbitals has radial nodes which increased the complexity in solving integrals in quantum chemical calculations.

Much more convenient are a class of modified orbitals called Slater-type orbitals (STOs). These differ from their hydrogenlike counterparts in that they have no radial nodes. Angular terms are identical in the two types of orbital. The unnormalized radial term for an STO is

Slater constructed rules for determining the values of s that would match the orbitals obtained from SCF calculation. These rules, appropriate for electrons up to the 3d level, are:The shielding constant s for an orbital associated with any of the above groups is the sum of the following contributions:(a)(b) 0.35 (except 0.30 in the 1s group).(c) s or p orbital,0.85, d orbital 1.00, (),s, p, or d orbitals, 1.00.

For example, N atom with ground state configuration 1s22s22p3, the 2s and 2p orbital would have the same radial part of STOs.Slater-type orbitals are very frequently used in quantum chemistry because they provide us with very good approximaiton to SCF atomic orbitals with almost no effort.

The STO have no radial nodes, so it loses some orthogonality, although the angular terms still give orthogonality between orbitals having different l or m quantum numbers. Therefore, STOs differing only in their n quantum number are nonorthogonal, such as 1s, 2s, 3s,.are nonorthogonal, 2pz,3pz, or, 3dxy, 4dxy, are nonorthogonal. Therefore, problem would arises if one forgets about its nonorthognality when making certain calculations. Aufbau principle (building up principle): the orbital ordering:1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 5d 4f 6p 7s 6d 5f .However, there is no fixing rule, it depends on the Z value of the atoms.

Explain briefly the observation that the energy difference between the 1s22s1 (2S1/2) state and 1s22p1 (2p1/2) state for Li is 14,904 cm-1, whereas for Li2+ the 2s1 (2S1/2) and 2p1(2p1/2) state are essentially degenerate. (They differ only by 2.4 cm-1).(hint: consider the hydrogen-like orbitals but not the Slator orbitals for the Li atom, the penetration of 2s is larger than the 2p, so the orbital energy of 2s is ? than 2p)

Combined spin-orbital angular momentum for one-electron ionsThe magnitude of this coupling angular momentum is

Russell Saunders Coupling Scheme (For non-equivalent electrons)()spin-orbit coupling is weak orbital momenta spin momenta Clebsch Gordan series :angular momentum

L > S, J term symbol multiplicity L < S for equivalent electrons, p2, (c: 1s22s22p2),, micro states,,, Lowe, p156, equivalent electrons.

Russell Saunders Coupling Scheme (spin-orbit coupling) j-j coupling

R S Coupling22 LSLSL+SJ j-j scheme J ?2

spin coupling

R-S coupling J

heavy atomR-S coupling j-j couplingtotal angular momentum (spin, orbit) j

total angular momentumcoupling J quantum number.R-S heavy atom term symbol j j

spin orbit coupling ?()spin magnetic moment heavy atom spin orbit coupling

Zeeman effectmagnetic moment ,

p orbital split,split 3

Terms where in J contains contributions from both L and S have Zeeman splittings other than one or two times the normal value, depending on the details of the way L or S are combined. The extent to which a term members energy is shifted by a magnetic field of strength B is Indicating that half of the z-component of angular momentum is due to the orbital motion, and half is due to spin (which is double weighted in its effect on magnetic moment).

electron spin magnetic moment Zeeman splitting, ESRElectron Spin Resonance) Nuclear spin magnetic momentZeeman splitting, NMRNuclear Magnetic Resonance)

Angular momentum for many-electron atoms (Equivalent electrons)

Molecular term symbolspoint group, point group character table representation,,,,, term symbol. H2Oterm symbol