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Many-electron atoms

Many-electron atoms.

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Many-electron atoms

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  1. 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):

  2. 其中左邊的1,2,3,…代表spatial coordinates of each of the n electrons. 因此1代表x1,y1,z1, or r1, θ1, ψ1,這裡我們就不再談原子本身的動能(視為是不動或固定能量),右邊第三項的寫法,保證 1/r12,和 1/r21不會同時出現,這樣兩電子的斥力才不會多算一次,也不會出現 1/r22等的 case 因此上式中,對 He atom 而言,其 hamiltonian 寫法為:

  3. is called one-electron operator, and 1/r12 is called two-electron operator.如果將後面這一項省略掉的話,那就是所謂粗略的近似法,不考慮電子間的排斥能,而只有個別獨立電子的能量。

  4. 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 fN for describing the electronic distribution about an atom. Products of the atomic orbitals Φi’s are eigenfNs of Happ, and the eigenvalue E is equal to the sum of the atomic orbital energies εi’s

  5. Simple products and electron exchange symmetry For the configuration 1s12s1, the wavefN 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 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.

  6. 欲達到此結果,可以有兩種情形,那就是將 1s(1)2s(2) and 1s(2)2s(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.

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

  8. 只有兩種spin,稱αand β,為電子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.

  9. Slater determinants Slater 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:

  10. For three electrons wavefNs, 1s22s1 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

  11. 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 2S+1=1，其中「 2S+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, 1s12s1,符合Pauli principle wavefNs:

  12. 如果用行列式的方式來表示時,發現無法用單一行列式來含蓋 triplet state。

  13. 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 wavefN.

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

  15. 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.

  16. Next we will investigate the energies of the states as they are described by these wavefNs We 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:

  17. 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:

  18. 動能部分:

  19. 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.

  20. 動能部分:

  21. so that this expansion becomes 位能的部分,expansion over(-2/r1, -2/r2),類似情形得

  22. 排斥能部分,1/r12, occurs in four two-electron integrals:

  23. 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,

  24. 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.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 fNs in the integrand differ by an exchange of electrons.

  25. K值代表the interaction between an electron ‘distribution’ described by 1s*2s, 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 larger than R, 此情形代表這兩電子 on opposite sides of the nodal surface, then 1s(1)2s(1)1s(2)2s(2) is negative. These positive or negative contributions to K are weighted by the fN of 1/r12 綜觀之,K值若大時,應為正值,若為負值應會是很小,可忽略。

  26. 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.

  27. What is the meaning of “Fermi hole” In 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.

  28. 如果是 singlet state (symmetric space fN) 當兩電子的 coordinate 相同時, 1s(1)=1s(2), wavefN 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wavefN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).

  29. 然而在 triplet state wavefN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 fN的設定是有很大的關係的,當basis fN越好時,其r12值卻越小,(參照Table 5-1,列出不同 wavefN的情況下,所計算出的結果,其 1/r12的平均值在 singlet and triplet states下有不同的趨勢。所以wavefN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 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.

  30. 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?

  31. 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, 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.

  32. 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 expandedorbital. 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.

  33. 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 consistentwith the potential due to the nucleus and the charge cloud for the other electron: the electrons move in a “self-consistent field” (SCF).

  34. 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.

  35. The hallmark(主要特徵) 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.

  36. 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

  37. 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)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 1s 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.

  38. 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.

  39. 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.

  40. 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)

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

  42. Russell – Saunders Coupling Scheme (For non-equivalent 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：

  43. 一般而言，若L > S, 則其 J 值個數與 term symbol 的multiplicity 相同，如上例的 但若 L < S，則就不依此法則，如上例 但 for equivalent electrons, 如 p2, (c: 1s22s22p2),就沒有那麼單純,須考慮各種 micro states,合適保留,不合適刪除, 參考 Lowe, p156, equivalent electrons.

  44. 一般Russell – Saunders Coupling Scheme 適用於較輕的原子 (其中電子本身spin-orbit coupling較小)，對於較重原子，就不適用，改為考慮每個原子間的 j-j coupling 。

  45. 在R –S Coupling中，如果有2個電子以上，則先算2個電子的 L，S，再和第三個電子重新做加成得到最終的L與S，再以L+S方式求出另J值；若適合 j-j scheme 再以另方式求出J值 ◦ 例如 電子組態為 ? 則先求出2個電子的

  46. 第三個電子 spin 的 coupling

  47. 由 R-S coupling 求J

  48. 如果是heavy atom時，R-S coupling不適用，必須用 j-j coupling。每一個電子只考慮total angular momentum (spin, orbit加成) j，每個電子再與每個電子的相互作用，此時的 就相對不太重要了。如

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