Manyelectron atoms In constructing the hamiltonian operator for

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Many-electron atoms In constructing the hamiltonian operator for a many electron atom, we shall

Many-electron atoms In 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):

Where n is the principal quantum number for atomic orbital Φi , and Z

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

Simple products and electron exchange symmetry For the configuration 1 s 12 s 1,

Simple products and electron exchange symmetry For the configuration 1 s 12 s 1, the wavef. N is : 因為 1和2電子無法分辨, 我們必須加以修正, so that it yields an average value for r 1 and r 2 that is independent of our choice of electron labels. This means that the electron density itself must be independent of our electron labeling scheme.

欲達到此結果, 可以有兩種情形, 那就是將 1 s(1)2 s(2) and 1 s(2)2 s(1) 相加, 或相減, 其平方後將1和2交換才 不會變。

欲達到此結果, 可以有兩種情形, 那就是將 1 s(1)2 s(2) and 1 s(2)2 s(1) 相加, 或相減, 其平方後將1和2交換才 不會變。 前式為 symmetric to the exchange of labels, 後式為 antisymmetric to the exchange of labels.

Electron spin and the exclusion principle Stern and Gerlach observed two bands of Ag

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

只有兩種spin, 稱αand β, 為電子normalized spin f. Ns, Pauli principle: wavef. Ns must be antisymmetric

只有兩種spin, 稱αand β, 為電子normalized spin f. Ns, Pauli principle: wavef. Ns must be antisymmetric with respect to simultaneous interchange of space and spin coordinates of electrons, called spin-orbitals of electrons.

Slater determinants Slater suggested that there is a simple way to write wavef. Ns

Slater determinants Slater suggested that there is a simple way to write wavef. Ns guaranteeing that they will be antisymmetric for interchange of electronic space and spin coordinates, for example 1 s 12 s 1:

For three electrons wavef. Ns, 1 s 22 s 1 寫法為先以行的方式將各電子的 spin-orbitals寫上去, 然後再 填入電子的

For three electrons wavef. Ns, 1 s 22 s 1 寫法為先以行的方式將各電子的 spin-orbitals寫上去, 然後再 填入電子的 indices, 第一行填 1, 第二行填 2, 第三行填 3。 展開後, 任意交換兩電子, 一定會變號, antisymmetric to the exchange of any two electron’s indices

Singlet and triplet states 有兩個電子填入同一個 orbital 時,必須是 paired (↑↓) ,其total spin, S為 0,則 2

Singlet and triplet states 有兩個電子填入同一個 orbital 時,必須是 paired (↑↓) ,其total spin, S為 0,則 2 S+1=1 ,其中「 2 S+1=1」 就稱為 spin-multiplicity 。 其值若為 1 叫 singlet,若為 2,叫「doublet」,若 為 3,叫「triplet」,4叫「quartet」…當然,若兩 個電子填入不同orbitals時,就可能為singlet 或是 triplet 。例如, excited state of He, 1 s 12 s 1, 符合Pauli principle wavef. Ns:

equivalent to (1) 展開式 equivalent to (2) and (4) The lesson to be learned

equivalent to (1) 展開式 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 wavef. N.

Paired spin, s=0, Ms=0, 但xy平面上 的分量仍然不斷變化 This is called Singlet.

Paired spin, s=0, Ms=0, 但xy平面上 的分量仍然不斷變化 This is called Singlet.

Two electrons with parallel spins, have a nonzero total spin angular momentum. 有三種方式,其中 the

Two electrons with parallel spins, have a nonzero total spin angular momentum. 有三種方式,其中 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. This is called triplet 與前面paired spin 比較: two paired spin are precisely antiparallel, however, two ‘parallel’ spins are not strictly parallel.

Next we will investigate the energies of the states as they are described by

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

We notice that the hamiltonian operator has no interaction term on spin part, this

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? 展開, 分為動能, 位能及排斥能三部分, 各別探討:

 • The orthogonality of the 1 s and 2 s orbitals caused the

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

so that this expansion becomes 位能的部分, expansion over (-2/r 1, -2/r 2), 類似情形得

so that this expansion becomes 位能的部分, expansion over (-2/r 1, -2/r 2), 類似情形得

排斥能部分, 1/r 12, occurs in four two-electron integrals:

排斥能部分, 1/r 12, occurs in four two-electron integrals:

Thus, the average energy value is: The first two terms gives the average energy

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

Where J and K represent the last two integrals. The integral J denotes electrons

Where J and K represent the last two integrals. The integral J denotes electrons 1 and 2 as being in ‘charge clouds’ described by 1 s*1 s and 2 s*2 s, respectively. The operator 1/r 12 gives the electrostatic repulsion energy between these two charge clouds. 因為這些是 電子雲的斥力, 所以J值是正的, 稱 coulomb integral. K is called an ‘exchange integral’ because the two product f. Ns in the integrand differ by an exchange of electrons.

K值代表the interaction between an electron ‘distribution’ described by 1 s*2 s, and another electron

K值代表the interaction between an electron ‘distribution’ described by 1 s*2 s, and another electron in the same distribution. (這些分部只是數學函數, 並非實質可畫出的分部 情形)。 當r 1 and r 2 are both smaller or Both larger, then the f. N 1 s(1)2 s(1)1 s(2)2 s(2) will be positive. But when one r value is smaller than R and the other is larger than R, 此情形代表這兩電子 on opposite sides of the nodal surface, then 1 s(1)2 s(1)1 s(2)2 s(2) is negative. These positive or negative contributions to K are weighted by the f. N of 1/r 12 綜觀之, K值若大時, 應為正值, 若為負值應會是很小, 可忽略。

 • Since the integral K is positive, we can see that from the

• 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 2 K.

What is the meaning of “Fermi hole” In triplet state the space part of

What is the meaning of “Fermi hole” In triplet state the space part of the wavef. N: What would happen if these two electrons are collide ? Which means that the coordinate of ‘ 1’ electron is equal to ‘ 2’ electron, that is, 1 s(1)=1 s(2), and 2 s(1)=2 s(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 wavef. N that is properly antisymmeterized.

如果是 singlet state (symmetric space f. N) 當兩電子的 coordinate 相同時, 1 s(1)=1 s(2), wavef.

如果是 singlet state (symmetric space f. N) 當兩電子的 coordinate 相同時, 1 s(1)=1 s(2), wavef. N 是否亦應該 vanish, (應與spin 無關才對), 這就是所謂的 coulomb hole. However, 在這個 wavef. N下卻沒看到 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).

然而在 triplet state wavef. N確實可由 Fermi hole 的存在, 而感受 到兩電子間的距離的確較長, 可是實際上的計算結果, 發現其與 basis f.

然而在 triplet state wavef. N確實可由 Fermi hole 的存在, 而感受 到兩電子間的距離的確較長, 可是實際上的計算結果, 發現其與 basis f. N的設定是有很大的關係的, 當basis f. N越好時, 其r 12值卻 越小, (參照Table 5 -1, 列出不同 wavef. N的情況下, 所計算出的結 果, 其 1/r 12的平均值在 singlet and triplet states下有不同的趨勢。 所以wavef. N的選擇有很重要的決定性)這說明了一個必須注意到 的現象, 那就是: Warning: Usable approximations to eigenf. Ns 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.

suppose we take ordinary independent-electron wavef. N as our initial approximation for the helium

suppose we take ordinary independent-electron wavef. N 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 Principle 一般作法是we can approximate this repulsion

The Self-Consistent Field, Slater-Type Orbitals, and the Aufbau Principle 一般作法是we can approximate this repulsion by saying that electron 1 ‘sees’ electron 2 as a smeared out, timeaveraged 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: [1 s(2)] 2.

Our approximation now has electron 1 moving in the field of a positive nucleus

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 ξ 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, ξwill 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

• 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 ξconverges to an unchanging value (under the constraint that electrons 1 and 2 ultimately occupy orbitals having the same value of ξ). 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 wavef. N in much closer accord

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

 • The hallmark(主要特徵) of independent electron treatment is a wavef. N containing only

• The hallmark(主要特徵) of independent electron treatment is a wavef. N containing only a product of one-electron f. Ns. There are no f. Ns of, say, r 12, which would make wavef. N depend on the instantaneous distance between electrons 1 and 2. • Atomic orbitals that are eigenf. Ns for the oneelectron 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

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

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 3 d level, are: (2)The shielding constant s for an orbital associated with any of the above groups is the sum of the following contributions: (a)比該電子更外層的電子不具遮蔽效應。 (b) 來自同層的每個電子遮蔽貢獻為 0. 35 (except 0. 30 in the 1 s group). (c) 來自內面一層的 s or p orbital, 每個電子的貢獻為 0. 85, d orbital 電子的貢獻為 1. 00, 來自內面更深一層(內面第二層) 以上的電子遮蔽貢獻, 不管s, p, or d orbitals, 每個電子的貢獻 皆為 1. 00.

For example, N atom with ground state configuration 1 s 22 p 3, the

For example, N atom with ground state configuration 1 s 22 p 3, the 2 s and 2 p 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

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 1 s, 2 s, 3 s, …. are nonorthogonal, 2 pz, 3 pz, … or, 3 dxy, 4 dxy, … are nonorthogonal. Therefore, problem would arises if one forgets about its nonorthognality when making certain calculations. Aufbau principle (building up principle): the orbital ordering: 1 s 2 s 2 p 3 s 3 p 4 s 3 d 4 p 5 s 4 d 5 p 6 s 5 d 4 f 6 p 7 s 6 d 5 f …. 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 1 s 22 s

Explain briefly the observation that the energy difference between the 1 s 22 s 1 (2 S 1/2) state and 1 s 22 p 1 (2 p 1/2) state for Li is 14, 904 cm-1, whereas for Li 2+ the 2 s 1 (2 S 1/2) and 2 p 1(2 p 1/2) 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 2 s is larger than the 2 p, so the orbital energy of 2 s is ? than 2 p)

Combined spin-orbital angular momentum for one-electron ions The magnitude of this coupling angular momentum

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

Russell – Saunders Coupling Scheme (For nonequivalent electrons)(適用於同量子數電子不在同一軌域) 用於多電子的spin-orbit coupling is weak,因此把多個電子 的 orbital

Russell – Saunders Coupling Scheme (For nonequivalent electrons)(適用於同量子數電子不在同一軌域) 用於多電子的spin-orbit coupling is weak,因此把多個電子 的 orbital momenta 一起合起來,再與多個電子合起來的 spin momenta 相互作用 Clebsch – Gordan series : 描述兩個angular momentum向量加 成的可能值,如:

第三個電子 spin 的 coupling

第三個電子 spin 的 coupling

由 R-S coupling 求J

由 R-S coupling 求J

 • Terms where in J contains contributions from both L and S have

• 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 member’s 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).

事實上 electron spin magnetic moment 亦會和外 加磁場作用產生Zeeman splitting, 例如ESR( Electron Spin Resonance) 就是利用此原理 。

事實上 electron spin magnetic moment 亦會和外 加磁場作用產生Zeeman splitting, 例如ESR( Electron Spin Resonance) 就是利用此原理 。 Nuclear spin magnetic moment亦會和外加磁場作用產生 Zeeman splitting, 例如NMR(Nuclear Magnetic Resonance) 就是利用此原理 。 但其 因為是質子,重量很大,所以作用能很小,所需要外加磁場 很大。

Angular momentum for manyelectron atoms (Equivalent electrons)

Angular momentum for manyelectron atoms (Equivalent electrons)