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The single-particle approximation

In the previous chapter we saw that except for the simplest solids, like those formed by noble elements or by purely ionic combinations which can be described essentially in classical terms, in all other cases we need to consider the behavior of the valence electrons. The following chapters deal with these valence electrons (we will also refer to them as simply "the electrons" in the solid); we will study how their behavior is influenced by, and in turn influences, the ions.
Our goal in this chapter is to establish the basis for the single-particle description of the valence electrons. We will do this by starting with the exact hamiltonian for the solid and introducing approximations in its solution, which lead to sets of single-particle equations for the electronic degrees of freedom in the external potential created by the presence of the ions. Each electron also experiences the presence of other electrons through an effective potential in the single-particle equations; this effective potential encapsulates the many-body nature of the true system in an approximate way. In the last section of this chapter we will provide a formal way for eliminating the core electrons from the picture, while keeping the important effect they have on valence electrons.

2.1 The hamiltonian of the solid
2.1 固體的哈密頓量

An exact theory for a system of ions and interacting electrons is inherently quantum mechanical, and is based on solving a many-body Schrödinger equation of the form
where is the hamiltonian of the system, containing the kinetic energy operators
系統的哈密頓量 ,包含動能算子
and the potential energy due to interactions between the ions and the electrons. In the above equations: is Planck's constant divided by is the mass of ion ;
由於離子和電子之間的相互作用而產生的潛在能量。在上述方程式中: 是普朗克常數除以 是離子 的質量;

is the mass of the electron; is the energy of the system; is the manybody wavefunction that describes the state of the system; are the positions of the ions; and are the variables that describe the electrons. Two electrons at repel one another, which produces a potential energy term
是電子的質量; 是系統的能量; 是描述系統狀態的多體波函數; 是離子的位置;而 是描述電子的變數。兩個位於 的電子會互相排斥,這將產生一個電位能項。
where is the electronic charge. An electron at is attracted to each positively charged ion at , producing a potential energy term
其中 是電子電荷。在 處的電子被吸引到每個在 處的正電荷離子,產生一個電位能項。
where is the valence charge of this ion (nucleus plus core electrons). The total external potential experienced by an electron due to the presence of the ions is
其中 是這個離子(核加上核心電子)的價電荷。由於離子的存在,電子所經歷的總外部電位是
Two ions at positions also repel one another giving rise to a potential energy term
兩個在位置 的離子也互相排斥,產生一個電位能項
Typically, we can think of the ions as moving slowly in space and the electrons responding instantaneously to any ionic motion, so that has an explicit dependence on the electronic degrees of freedom alone: this is known as the Born-Oppenheimer approximation. Its validity is based on the huge difference of mass between ions and electrons (three to five orders of magnitude), making the former behave like classical particles. The only exception to this, noted in the previous chapter, are the lightest elements (especially ), where the ions have to be treated as quantum mechanical particles. We can then omit the quantum mechanical term for the kinetic energy of the ions, and take their kinetic energy into account as a classical contribution. If the ions are at rest, the hamiltonian of the system becomes
通常,我們可以將離子視為在空間中緩慢移動,而電子對任何離子運動的即時響應,因此 僅對電子自由度有明確依賴:這被稱為波恩-奧本海默近似。其有效性基於離子和電子之間的質量巨大差異(三到五個數量級),使前者表現得像經典粒子。唯一的例外是在先前章節中指出的最輕元素(特別是 ),在這裡,離子必須被視為量子機械粒子。然後,我們可以省略離子動能的量子機械項,並將其動能視為經典貢獻。如果離子靜止,系統的哈密頓量就變成
In the following we will neglect for the moment the last term, which as far as the electron degrees of freedom are concerned is simply a constant. We discuss how this constant can be calculated for crystals in Appendix F (you will recognize in this term the Madelung energy of the ions, mentioned in chapter 1). The hamiltonian then takes the form
在接下來的內容中,我們暫時忽略最後一項,就電子自由度而言,這只是一個常數。我們將在附錄 F 中討論如何計算晶體的這個常數(您將在這個項中認出離子的 Madelung 能量,這在第 1 章中提到)。哈密頓量的形式如下:
with the ionic potential that every electron experiences defined in Eq. (2.5).
具有每個電子所經歷的離子勢能 ,在方程式(2.5)中定義。
Even with this simplification, however, solving for is an extremely difficult task, because of the nature of the electrons. If two electrons of the same spin interchange positions, must change sign; this is known as the "exchange" property, and is a manifestation of the Pauli exclusion principle. Moreover, each electron is affected by the motion of every other electron in the system; this is known as the "correlation" property. It is possible to produce a simpler, approximate picture, in which we describe the system as a collection of classical ions and essentially single quantum mechanical particles that reproduce the behavior of the electrons: this is the single-particle picture. It is an appropriate description when the effects of exchange and correlation are not crucial for describing the phenomena we are interested in. Such phenomena include, for example, optical excitations in solids, the conduction of electricity in the usual ohmic manner, and all properties of solids that have to do with cohesion (such as mechanical properties). Phenomena which are outside the scope of the single-particle picture include all the situations where electron exchange and correlation effects are crucial, such as superconductivity, transport in high magnetic fields (the quantum Hall effects), etc.
即使進行了這種簡化,解決 的問題仍然是一項極其困難的任務,這是由於電子的性質。如果兩個自旋相同的電子交換位置, 必須改變符號;這被稱為“交換”性質,是 Pauli 排斥原則的表現。此外,每個電子都受到系統中每個其他電子運動的影響;這被稱為“相關”性質。我們可以產生一個更簡單的、近似的圖像,其中我們將系統描述為一組經典離子和基本上是重現電子行為的單個量子機械粒子:這就是單粒子圖像。當交換和相關效應對於描述我們感興趣的現象不是至關重要時,這是一個適當的描述。這些現象包括固體中的光學激發、通常的歐姆方式導電以及與凝聚力有關的固體所有性質(如機械性質)。 單粒子圖像範圍之外的現象包括所有電子交換和相關效應至關重要的情況,如超導性、在高磁場中的傳輸(量子霍爾效應)等。
In developing the one-electron picture of solids, we will not neglect the exchange and correlation effects between electrons, we will simply take them into account in an average way; this is often referred to as a mean-field approximation for the electron-electron interactions. To do this, we have to pass from the many-body picture to an equivalent one-electron picture. We will first derive equations that look like single-particle equations, and then try to explore their meaning.

2.2 The Hartree and Hartree-Fock approximations
2.2 哈特里和哈特里-福克近似。

2.2.1 The Hartree approximation
2.2.1 哈特里近似

The simplest approach is to assume a specific form for the many-body wavefunction which would be appropriate if the electrons were non-interacting particles, namely
with the index running over all electrons. The wavefunctions are states in which the individual electrons would be if this were a realistic approximation. These are single-particle states, normalized to unity. This is known as the Hartree approximation (hence the superscript ). With this approximation, the total energy of the system becomes
隨著指數 遍歷所有電子。波函數 是個體電子所在的狀態,如果這是一個現實的近似值。這些是單粒子狀態,歸一化為單位。這被稱為哈特里近似(因此上標 )。通過這種近似,系統的總能量變為
Using a variational argument, we obtain from this the single-particle Hartree equations:
where the constants are Lagrange multipliers introduced to take into account the normalization of the single-particle states (the bra and ket notation for single-particle states and its extension to many-particle states constructed as products of single-particle states is discussed in Appendix B). Each orbital can then be determined by solving the corresponding single-particle Schrödinger equation, if all the other orbitals were known. In principle, this problem of self-consistency, i.e. the fact that the equation for one depends on all the other 's, can be solved iteratively. We assume a set of 's, use these to construct the single-particle hamiltonian, which allows us to solve the equations for each new ; we then compare the resulting 's with the original ones, and modify the original 's so that they resemble more the new 's. This cycle is continued until input and output 's are the same up to a tolerance , as illustrated in Fig. 2.1 (in this example, the comparison of input and output wavefunctions is made through the densities, as would be natural in Density Functional Theory, discussed below).
常數 是拉格朗日乘數,用於考慮單粒子狀態 的正規化(單粒子狀態的 bra 和 ket 符號以及構建為單粒子狀態乘積的多粒子狀態在附錄 B 中討論)。然後,每個軌道 可以通過解決相應的單粒子薛定莊方程式來確定,如果所有其他軌道 都已知。原則上,這個自洽性問題,即一個方程式取決於所有其他 的事實,可以通過迭代解決。我們假設一組 ,使用這些來構建單粒子哈密頓量,這使我們能夠解決每個新 的方程式;然後我們將結果 與原始的進行比較,並修改原始的 ,使其更像新的 。這個循環將持續進行,直到輸入和輸出 在容差 內相同,如圖 2 所示。在這個例子中,通過密度比較輸入和輸出波函數,就像在下面討論的密度泛函理論中那樣自然。
The more important problem is to determine how realistic the solution is. We can make the original trial 's orthogonal, and maintain the orthogonality at each cycle of the self-consistency iteration to make sure the final 's are also orthogonal. Then we would have a set of orbitals that would look like single particles, each experiencing the ionic potential as well as a potential due to the presence of all other electrons, given by
更重要的問題是確定解決方案的現實性。我們可以使原始試驗 正交,並在自洽迭代的每個週期中保持正交性,以確保最絈 也是正交的。然後我們將擁有一組看起來像單個粒子的軌道,每個 都會感受到離子勢 以及由所有其他電子存在引起的勢
Figure 2.1. Schematic representation of iterative solution of coupled single-particle equations. This kind of operation is easily implemented on the computer.
圖 2.1. 耦合單粒子方程迭代解的示意圖。這種操作在電腦上很容易實現。
This is known as the Hartree potential and includes only the Coulomb repulsion between electrons. The potential is different for each particle. It is a mean-field approximation to the electron-electron interaction, taking into account the electronic charge only, which is a severe simplification.

2.2.2 Example of a variational calculation
2.2.2 變分計算的例子

We will demonstrate the variational derivation of single-particle states in the case of the Hartree approximation, where the energy is given by Eq. (2.10), starting with the many-body wavefunction of Eq. (2.9). We assume that this state is a stationary state of the system, so that any variation in the wavefunction will give a zero variation in the energy (this is equivalent to the statement that the derivative of a function at an extremum is zero). We can take the variation in the wavefunction to be of the form , subject to the constraint that , which can be taken into account by introducing a Lagrange multiplier :
我們將展示在 Hartree 近似情況下單粒子狀態的變分導出,其中能量由方程式 (2.10) 給出,從方程式 (2.9) 的多體波函數開始。我們假設這個狀態是系統的定態,因此波函數的任何變化將導致能量的變化為零(這相當於說一個函數在極值點的導數為零)。我們可以假設波函數的變化形式為 ,受到約束條件 的限制,這可以通過引入拉格朗日乘數 來考慮:
Notice that the variations of the bra and the ket of are considered to be independent of each other; this is allowed because the wavefunctions are complex quantities, so varying the bra and the ket independently is equivalent to varying the real and
請注意, 的 bra 和 ket 的變化被認為是彼此獨立的;這是允許的,因為波函數是複數,所以獨立變化 bra 和 ket 相當於變化實部和

imaginary parts of a complex variable independently, which is legitimate since they represent independent components (for a more detailed justification of this see, for example, Ref. [15]). The above variation then produces
Since this has to be true for any variation , we conclude that
由於這對於任何變化 都必須是真實的,我們得出結論
which is the Hartree single-particle equation, Eq. (2.11).

2.2.3 The Hartree-Fock approximation
2.2.3 哈特里-福克近似法

The next level of sophistication is to try to incorporate the fermionic nature of electrons in the many-body wavefunction . To this end, we can choose a wavefunction which is a properly antisymmetrized version of the Hartree wavefunction, that is, it changes sign when the coordinates of two electrons are interchanged. This is known as the Hartree-Fock approximation. For simplicity we will neglect the spin of electrons and keep only the spatial degrees of freedom. This does not imply any serious restriction; in fact, at the Hartree-Fock level it is a simple matter to include explicitly the spin degrees of freedom, by considering electrons with up and down spins at position . Combining then Hartree-type wavefunctions to form a properly antisymmetrized wavefunction for the system, we obtain the determinant (first introduced by Slater [16]):
下一個更複雜的層次是嘗試將電子的費米特性納入多體波函數 中。為此,我們可以選擇一個適當反對稱化的哈特里波函數版本,即當兩個電子的坐標互換時,它會改變符號。這被稱為哈特里-福克近似。為了簡化,我們將忽略電子的自旋,僅保留空間自由度。這並不意味著任何嚴重的限制;事實上,在哈特里-福克級別上,通過考慮位置 處具有上旋和下旋的電子,可以輕鬆地明確包含自旋自由度。然後將哈特里型波函數組合成系統的適當反對稱化波函數,我們獲得行列式(由斯萊特[16]首次引入):
where is the total number of electrons. This has the desired property, since interchanging the position of two electrons is equivalent to interchanging the corresponding columns in the determinant, which changes its sign.
其中 是電子的總數。這具有所需的特性,因為交換兩個電子的位置等同於交換行列式中對應的列,這將改變其符號。
The total energy with the Hartree-Fock wavefunction is
Hartree-Fock 波函數的總能量是