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. 在前一章中，我們看到除了最簡單的固體外，如由貴族元素形成的固體或純粹離子組合，這些固體基本上可以用古典術語來描述，對於所有其他情況，我們需要考慮價電子的行為。接下來的章節將討論這些價電子（我們也將簡單地稱之為固體中的“電子”）；我們將研究它們的行為如何受到影響，並反過來影響離子。
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. 本章的目標是建立價電子的單粒子描述基礎。我們將從固體的精確哈密頓量開始，並在其解中引入近似，從而導致在由離子存在所創造的外部電位中的電子自由度的單粒子方程組。每個電子也通過單粒子方程中的有效電位感受其他電子的存在；這個有效電位以近似方式封裝了真實系統的多體性質。在本章的最後一節中，我們將提供一種正式的方法來消除核心電子的影響，同時保留它們對價電子的重要影響。
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 一個關於離子系統和相互作用電子的確切理論在本質上是量子力學的，並且基於解決一個形式為多體薛丁格方程的問題
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. 在發展固體的單電子模型時，我們不會忽略電子之間的交換和相關效應，我們只是以平均方式考慮它們；這通常被稱為對電子-電子相互作用的平均場近似。為了做到這一點，我們必須從多體圖像轉換為等效的單電子圖像。我們將首先推導出看起來像單粒子方程的方程式，然後嘗試探索它們的含義。
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 最簡單的方法是假設一個特定形式的多體波函數，如果電子是非相互作用的粒子，這將是適當的
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: 使用變分論證，我們從中得到單粒子哈特里方程式：
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. 這被稱為哈特里電位，僅包括電子之間的庫倫排斥。對於每個粒子，電位是不同的。這是對電子-電子相互作用的平均場近似，僅考慮電荷，這是一種嚴重的簡化。
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 複變數的虛部獨立地，這是合法的，因為它們代表獨立的分量（有關此的更詳細理由，請參見，例如，參考文獻[15]）。然後上述變化產生
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). 這是哈特里單粒子方程式，方程式（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
and the single-particle Hartree-Fock equations , obtained by a variational calculation, are 通過變分計算獲得的單粒子哈特里-福克方程
This equation has one extra term compared with the Hartree equation, the last one, which is called the "exchange" term. The exchange term describes the effects of exchange between electrons, which we put in the Hartree-Fock many-particle wavefunction by construction. This term has the peculiar character that it cannot be written simply as (in the following we use the superscript to denote "exchange"). It is instructive to try to put this term in such a form, by multiplying and dividing by the proper factors. First we express the Hartree term in a different way: define the single-particle and the total densities as 這個方程式與哈特里方程式相比多了一個額外項，即最後一項，稱為「交換」項。交換項描述了電子之間的交換效應，我們通過構造將其放入哈特里-福克多粒子波函數中。這個項具有特殊的特性，不能簡單地寫成 （以下我們使用上標 表示「交換」）。嘗試將這個項以適當的因子相乘和除法的方式表達是有益的。首先，我們以不同的方式表達哈特里項：定義單粒子和總密度為
so that the Hartree potential takes the form 使得哈特里電位呈現如下形式
Now construct the single-particle exchange density to be 現在構建單粒子交換密度為
Then the single-particle Hartree-Fock equations take the form 然後單粒子哈特里-福克方程式的形式為
with the exchange potential, in analogy with the Hartree potential, given by 具有交換電位，與哈特里電位類比給定
The Hartree and exchange potentials give the following potential for electronelectron interaction in the Hartree-Fock approximation: Hartree 和交換勢給出了 Hartree-Fock 近似中電子間相互作用的潛力如下：
which can be written, with the help of the Hartree-Fock density 可以借助哈特里-福克密度來寫
as the following expression for the total electron-electron interaction potential: 作為總電子-電子相互作用勢的下列表達式：
The first term is the total Coulomb repulsion potential of electrons common for all states , while the second term is the effect of fermionic exchange, and is different for each state . 第一項是對所有狀態共同的電子庫倫排斥勢能，而第二項是費米子交換的效應，對每個狀態都不同。
2.3 Hartree-Fock theory of free electrons 2.3 自由電子的哈特里-福克理論
To elucidate the physical meaning of the approximations introduced above we will consider the simplest possible case, that is one in which the ionic potential is a uniformly distributed positive background. This is referred to as the jellium model. In this case, the electronic states must also reflect this symmetry of the potential, which is uniform, so they must be plane waves: 為了闡明上述引入的近似值的物理意義，我們將考慮最簡單的情況，即離子勢是均勻分佈的正背景。這被稱為凝膠模型。在這種情況下，電子狀態也必須反映出勢的對稱性，即均勻，因此它們必須是平面波：
where is the volume of the solid and is the wave-vector which characterizes state . Since the wave-vectors suffice to characterize the single-particle states, we will use those as the only index, i.e. . Plane waves are actually a very convenient and useful basis for expressing various physical quantities. In particular, they allow the use of Fourier transform techniques, which simplify the calculations. In the following we will be using relations implied by the Fourier transform method, which are proven in Appendix G. 其中 是固體的體積， 是表徵狀態 的波矢。由於波矢足以表徵單粒子狀態，我們將僅使用它們作為唯一的索引，即 。平面波實際上是表達各種物理量非常方便和有用的基礎。特別是，它們允許使用傅立葉變換技術，這簡化了計算。在接下來的內容中，我們將使用傅立葉變換方法隱含的關係，這些關係在附錄 G 中得到證明。
We also define certain useful quantities related to the density of the uniform electron gas: the wave-vectors have a range of values from zero up to some maximum 我們還定義了與均勻電子氣密度相關的某些有用量：波矢的值範圍從零到某個最大值
magnitude , the Fermi momentum, which is related to the density through 幅度 ，費米動量，與密度 有關
(see Appendix D, Eq. (D.10)). The Fermi energy is given in terms of the Fermi momentum (見附錄 D，方程式(D.10))。費米能量以費米動量表示。
It is often useful to express equations in terms of another quantity, , which is defined as the radius of the sphere whose volume corresponds to the average volume per electron: 通常將方程式表示為另一個量 是有用的，該量被定義為球的半徑，其體積對應於每個電子的平均體積：
and is typically measured in atomic units (the Bohr radius, ). This gives the following expression for : 通常以原子單位（玻爾半徑， ）來衡量 。這給出了 的以下表達式：
where the last expression contains the dimensionless combinations of variables and . If the electrons had only kinetic energy, the total energy of the system would be given by 最後一個表達式包含變量 和 的無因次組合。如果電子只有動能，系統的總能量將由以下公式給出:
(see Appendix D, Eq. (D.12)). Finally, we introduce the unit of energy rydberg (Ry), which is the natural unit for energies in solids, (見附錄 D，方程式(D.12))。最後，我們介紹了能量單位里德堡（Ry），這是固體能量的自然單位，
With the electrons represented by plane waves, the electronic density must be uniform and equal to the ionic density. These two terms, the uniform positive ionic charge and the uniform negative electronic charge of equal density, cancel each other. The only terms remaining in the single-particle equation are the kinetic energy and the part of corresponding to exchange, which arises from : 以平面波表示的電子，電子密度必須是均勻的，並且等於離子密度。這兩個術語，均勻的正離子電荷和均勻的負電子電荷相等的密度，互相抵消。在單粒子方程式中剩下的唯一術語是動能和對應於交換的部分，這部分來自：
We have asserted above that the behavior of electrons in this system is described by plane waves; we prove this statement next. Plane waves are of course eigenfunctions of the kinetic energy operator: 我們已經斷言過，這個系統中電子的行為是由平面波描述的；我們接下來證明這個說法。平面波當然是動能算子的特徵函數：
so that all we need to show is that they are also eigenfunctions of the second term in the hamiltonian of Eq. (2.33). Using Eq. (2.24) for we obtain 這樣我們只需要證明它們也是方程式（2.33）中哈密頓量的第二項的特徵函數。使用方程式（2.24）對 ，我們得到
Expressing in terms of its Fourier transform provides a convenient way for evaluating the last sum once it has been turned into an integral using Eq. (D.8). The inverse Fourier transform of turns out to be 將 表示為其傅立葉變換的形式，提供了一種方便的方法，用來評估最後的總和，一旦它被轉換為使用等式(D.8)的積分。 的傅立葉逆變換結果是
as proven in Appendix G. Substituting this expression into the previous equation we obtain 如附錄 G 所證明。將這個表達式代入先前的方程式，我們得到
At this point it will be necessary to employ the Fourier transform representation of the -function, which is derived in Appendix G; this representation allows us to identify the quantity in square brackets in the last expression with a -function in momentum space, 在這一點上，將需要使用 -function 的傅立葉變換表示，該表示是在附錄 G 中推導出來的；這種表示使我們能夠將最後表達式中的方括號內的數量識別為動量空間中的 -function
which upon integration over gives 將其積分後得到
Figure 2.2. Energy (in rydbergs) of individual single-particle states as a function of momentum (with Fermi momentum), as given by Eq. (2.41), for two different values of (in ). The dashed curves give the kinetic energy contribution (first term on the right-hand side of Eq. (2.41)). 圖 2.2. 單粒子狀態的能量（以 Rydberg 為單位）作為動量 （以 Fermi 動量為單位）的函數，根據方程式（2.41）給出，對應兩個不同 值（以 為單位）。虛線曲線給出了動能貢獻（方程式（2.41）右側的第一項）。
with , where the function is defined as 與 ，其中函數 被定義為
This completes the proof that plane waves are eigenfunctions of the single-particle hamiltonian in Eq. (2.33). 這完成了證明平面波是單粒子哈密頓量在方程（2.33）中的特徵函數的證明。
With this result, the energy of single-particle state is given by 根據這個結果，單粒子狀態 的能量為
which, using the variable introduced in Eq. (2.29) and the definition of the Ry for the energy unit given in Eq. (2.32), can be rewritten in the following form: 使用在等式（2.29）中引入的變數 和在等式（2.32）中給出的能量單位 Ry 的定義，可以重寫為以下形式：
The behavior of the energy as function of the momentum (in units of ) is illustrated in Fig. 2.2 for two different values of . 能量 作為動量的函數（以 為單位）的行為在圖 2.2 中以兩個不同的 值來說明。
This is an intriguing result: it shows that, even though plane waves are eigenstates of this hypothetical system, due to the exchange interaction the energy of state 這是一個有趣的結果：它顯示，即使平面波是這個假想系統的本徵態，由於交換作用，狀態 的能量也會受到影響
is not simply , as might be expected for non-interacting particles; it also contains the term proportional to in Eq. (2.40). This term has interesting behavior at , as is evident in Fig. 2.2. It also gives a lower energy than the non-interacting electron case for all values of , an effect which is more pronounced for small values of (see Fig. 2.2). Thus, the electron-electron interaction included at the Hartree-Fock level lowers the energy of the system significantly. We can calculate the total energy of this system by summing the single-particle energies over up to momentum : 不僅僅是 ，這可能是非相互作用粒子所期望的；在方程式（2.40）中還包含與 成比例的項。這個項在 有有趣的行為，如圖 2.2 所示。對於所有 的值，它還給出比非相互作用電子情況更低的能量，這種效應在 的小值時更加明顯（見圖 2.2）。因此，包含在 Hartree-Fock 水平的電子-電子相互作用顯著降低了系統的能量。我們可以通過將單粒子能量相加到 的動量，計算這個系統的總能量：
Notice that we must include a factor of 2 for the spin of the electrons in both summations. This is indeed explicitly done for the kinetic energy part (see Appendix D where the expression for was derived). But for the second term, which represents the effective electron-electron interaction due to exchange, this factor of 2 is canceled by a factor of needed to compensate double counting of the effective interaction in the sum of 's: remember that this effective interaction is contained in the single-particle equations Eq. (2.16) as the sum over all states other than state , so if we simply sum all these contributions contained in the 's we will be counting each contribution twice. Turning the second term in the above equation into an integral through the usual procedure, we can evaluate the sum to find 請注意，在這兩個總和中，我們必須包括電子自旋的因子 2。這確實是對動能部分明確完成的（請參見附錄 D，那裡推導出 的表達式）。但對於第二項，代表由於交換而產生的有效電子-電子相互作用，這個因子 2 被需要來補償在 的總和中對有效相互作用的重複計算的因子 所抵消：請記住，這種有效相互作用包含在 單粒子方程式 Eq.（2.16）中，作為除了狀態 之外的所有狀態的總和，因此如果我們簡單地將包含在 中的所有這些貢獻相加，我們將計算每個貢獻兩次。通過常規程序將上述方程式的第二項轉換為積分，我們可以評估總和以找到
which quantifies by how much the effective electron-electron interaction due to exchange lowers the energy of the system relative to the kinetic energy alone. Using the expression of in terms of , Eq. (2.30), and expressing everything in rydbergs with the help of Eq. (2.32), we obtain 這量化了由於交換而產生的有效電子-電子相互作用降低系統能量相對於僅動能的程度。使用 的表達式 ，等式(2.30)，並借助等式(2.32)將所有內容表示為裡德堡，我們得到
This result should be compared with the expansion for the exact energy of the electron gas in the high-density limit (low values), first obtained by GellMann and Brueckner (for details and original references see Ref. [17]), 這個結果應該與電子氣體的精確能量在高密度極限（低 值）的擴展進行比較，這是由 GellMann 和 Brueckner 首次獲得的（有關詳細信息和原始參考請參見參考文獻[17]）。
It is quite remarkable that the Hartree-Fock approximation, based on an ad hoc expression for the many-body wavefunction, captures the first two terms in the 這是相當引人注目的，哈特里-福克近似是基於對多體波函數的一個臨時表達，捕捉了前兩項
exact expansion of the total energy. Of course in real situations this may not be very helpful, since in typical metals varies between 2 and 6 . 總能量的精確擴展。當然，在實際情況下，這可能並不是非常有幫助，因為在典型金屬中 在 2 和 6 之間變化。
Another interesting point is that we can express the potential due to exchange in a way that involves the density. This potential will give rise to the second term on the right-hand side of Eq. (2.44), namely 另一個有趣的觀點是，我們可以表達由於交換而產生的潛力，並涉及密度。 這個潛力將導致方程式（2.44）右側的第二項，即
which, using the expressions for discussed earlier, can be written as 使用先前討論的 的表達式，可以寫成
One of the most insightful proposals in the early calculations of the properties of solids, due to Slater [18], was to generalize this term for situations where the density is not constant, that is, a system with non-homogeneous distribution of electrons. In this case the exchange energy would arise from a potential energy term in the single-particle hamiltonian which will have the form 在固體性質的早期計算中，由斯萊特[18]提出的最具洞察力的建議之一是將這個術語概括為密度不恆定的情況，即具有非均勻分佈電子的系統。在這種情況下，交換能將源自單粒子哈密頓量中的一個潛在能項，其形式將為
where an extra factor of 2 is introduced to account for the fact that a variational derivation gives rise to a potential term in the single-particle equations which is twice as large as the corresponding energy term; conversely, when one calculates the total energy by summing terms in the single-particle equations, a factor of must be introduced to account for double-counting of interactions. In the last equation, the density, and hence the Fermi momentum, have become functions of , i.e. they can be non-homogeneous. There is actually good justification to use such a term in single-particle equations in order to describe the exchange contribution, although the values of the constants involved are different from Slater's (see also section 2.5 on Density Functional Theory). 在進行變分推導時，引入了一個額外的因子 2，以考慮到變分導出會產生單粒子方程中的一個電位項，其大小是對應能量項的兩倍；相反地，當通過將單粒子方程中的項相加來計算總能量時，必須引入一個因子 0，以考慮到相互作用的重複計算。在最後一個方程式中，密度和因此費米動量已經成為 的函數，即它們可以是非均勻的。實際上，有很好的理由在單粒子方程中使用這樣的項來描述交換貢獻，儘管所涉及的常數值與斯萊特（另見密度泛函理論第 2.5 節）的不同。
2.4 The hydrogen molecule 2.4 氫分子
In order to demonstrate the difficulty of including explicitly all the interaction effects in a system with more than one electron, we discuss briefly a model of the hydrogen molecule. This molecule consists of two protons and two electrons, so it is the simplest possible system for studying electron-electron interactions in a 為了展示在一個具有多個電子的系統中明確包含所有交互作用效應的困難，我們簡要討論了氫分子的模型。這個分子由兩個質子和兩個電子組成，因此它是研究電子間相互作用的最簡單系統。
realistic manner. We begin by defining the hamiltonian for a single hydrogen atom: 以實際方式。 我們首先定義單個氫原子的哈密頓量：
where is the position of the first proton. The wavefunction for this hamiltonian is , the ground state of the hydrogen atom with energy . Similarly, an atom at position (far from ) will have the hamiltonian 其中 是第一個質子的位置。這個哈密頓量的波函數是 ，氫原子的基態能量為 。同樣地，位於位置 （遠離 ）的原子將具有哈密頓量。
and the wavefunction . When the two protons are very far away, the two electrons do not interact and the two electronic wavefunctions are the same, only centered at different points in space. When the atoms are brought together, the new hamiltonian becomes 當兩個質子相距很遠時，兩個電子不會互動，兩個電子波函數相同，只是在空間中不同的點上居中。當原子被聚集在一起時，新的哈密頓變成
where the last four terms represent electron-proton attraction between the electron in one atom and the proton in the other (the cross terms), and electron-electron, and proton-proton repulsion. As we have done so far, we will ignore the protonproton repulsion, (last term in Eq. (2.51)), since it is only a constant term as far as the electrons are concerned, and it does not change the character of the electronic wavefunction. This is equivalent to applying the Born-Oppenheimer approximation to the problem and neglecting the quantum nature of the protons, even though we mentioned in chapter 1 that this may not be appropriate for hydrogen. The justification for using this approximation here is that we are concentrating our attention on the electron-electron interactions in the simplest possible model rather than attempting to give a realistic picture of the system as a whole. Solving for the wavefunction of this new hamiltonian analytically is already an impossible task. We will attempt to do this approximately, using the orbitals and as a convenient basis. 最後四項代表一個原子中的電子與另一個原子中的質子之間的電子-質子吸引力（交叉項），以及電子-電子和質子-質子斥力。正如我們迄今所做的那樣，我們將忽略質子-質子斥力（方程式（2.51）中的最後一項），因為就電子而言，它只是一個常數項，並且不會改變電子波函數的特性。這相當於將伯恩-奧本海默近似應用於問題並忽略質子的量子性質，即使我們在第 1 章中提到這對氫來說可能不適用。在這裡使用這種近似的理由是，我們專注於最簡單的模型中的電子-電子相互作用，而不是試圖給出系統整體的現實圖像。解析這個新哈密頓量的波函數 已經是一個不可能的任務。我們將嘗試以軌道 和 作為方便的基礎來近似解決這個問題。
If we were dealing with a single electron, then this electron would see the following hamiltonian, in the presence of the two protons: 如果我們正在處理一個單個電子，那麼這個電子將在兩個質子存在的情況下看到以下的哈密頓量：
We define the expectation value of this hamiltonian in the state or to be 我們將這個哈密頓量在狀態 或 中的期望值定義為
Notice that or are not eigenstates of , and . Also notice that we can write the total hamiltonian as 請注意， 或 不是 和 的特徵態。同時請注意，我們可以將總哈密頓寫成
We call the very last term in this expression, the electron-electron repulsion, the "interaction" term. It will prove convenient within the basis to define the so called "hopping" matrix elements 我們稱這個表達式中的最後一項，即電子-電子斥力，為“交互”項。在 基礎中，定義所謂的“跳躍”矩陣元素將證明是方便的。
where we can choose the phases in the wavefunctions to make sure that is a real positive number. These matrix elements describe the probability of one electron "hopping" from state to (or vice versa), within the singleparticle hamiltonian . A different term we can define is the "on-site" repulsive interaction between two electrons, which arises from the interaction term when the two electrons are placed at the same orbital: 我們可以選擇波函數中的相位 ，以確保 是一個實數正數。這些矩陣元素描述了一個電子從狀態 跳躍到 （或反之亦然）的概率，這是在單粒子哈密頓量 中的。我們可以定義的另一個術語是兩個電子之間的“現場”斥力相互作用，這是當兩個電子放置在同一軌道時產生的相互作用項：
where is also taken to be a real positive quantity. A model based on these physical quantities, the hopping matrix element and the on-site Coulomb repulsion energy, was introduced originally by Hubbard [18-20]. The model contains the bare essentials for describing electron-electron interactions in solids, and has found many applications, especially in highly correlated electron systems. Despite its apparent simplicity, the Hubbard model has not been solved analytically, and research continues today to try to understand its physics. 其中 也被視為一個實際正數量。一個基於這些物理量、跳躍矩陣元素和現場庫倫斥力能量的模型最初由哈伯德[18-20]引入。這個模型包含描述固體中電子-電子相互作用的基本要素，並在高度相關的電子系統中找到了許多應用。儘管哈伯德模型表面上看起來很簡單，但尚未被解析解，研究仍在繼續，以試圖理解其物理學。
Now we want to construct single-particle orbitals for , using as a basis and , which reflect the basic symmetry of the hamiltonian, that is, inversion relative to the midpoint of the distance between the two protons (the center of the molecule). There are two such possibilities: 現在我們想要構建單粒子軌道，使用 作為基礎，反映哈密頓尼安的基本對稱性，即相對於兩個質子之間距離的中點（分子的中心）的反轉。有兩種可能性：
the first being a symmetric and the second an antisymmetric wavefunction, upon inversion with respect to the center of the molecule; these are illustrated in Fig. 2.3. 第一個是對於分子中心進行反轉的對稱波函數，第二個是反對稱波函數；這些在圖 2.3 中有所說明。
Figure 2.3. Schematic representation of the hydrogen wavefunctions for isolated atoms (top panel), and the linear combinations that preserve the inversion symmetry with respect to the center of the molecule: a symmetric combination (bottom panel, left) and an antisymmetric combination (bottom panel, right). The latter two are the states defined in Eq. (2.57). 圖 2.3. 孤立原子的氫波函數的示意圖（頂部面板），以及保持相對於分子中心的反演對稱性的線性組合：對稱組合（底部面板，左）和反對稱組合（底部面板，右）。後兩者是方程式（2.57）中定義的狀態。
The expectation values of in terms of the are 在 方面的 的期望值是
Using the single-particle orbitals of Eq. (2.57), we can write three possible Hartree wavefunctions: 使用方程式(2.57)的單粒子軌道，我們可以寫出三個可能的哈特里波函數：
Notice that both and place the two electrons in the same state; this is allowed because of the electron spin. The expectation values of the two-particle hamiltonian which contains the interaction term, in terms of the , are 請注意， 和 都將兩個電子放在相同的狀態中；這是因為電子自旋的緣故。包含交互作用項的雙粒子哈密頓量 的期望值，以 表示，為
Next we try to construct the Hartree-Fock approximation for this problem. We will assume that the total wavefunction has a spin-singlet part, that is, the spin degrees of freedom of the electrons form a totally antisymmetric combination, 接下來我們嘗試為這個問題建構哈特里-福克近似。我們將假設總波函數具有一個自旋單重子部分，也就是說，電子的自旋自由度形成一個完全反對稱的組合。
which multiplies the spatial part of the wavefunction. This tells us that the spatial part of the wavefunction should be totally symmetric. One possible choice is 將波函數的空間部分乘以。這告訴我們波函數的空間部分應該是完全對稱的。一種可能的選擇是
The wavefunction is known as the Heitler-London approximation. Two other possible choices for totally symmetric spatial wavefunctions are 波函數 被稱為海特勒-倫敦近似。另外兩種可能的選擇是完全對稱的空間波函數。
Using the three functions , we can construct matrix elements of the hamiltonian and we can diagonalize this matrix to find its eigenvalues and eigenstates. This exercise shows that the ground state energy is 使用三個函數 ，我們可以構造哈密頓矩陣 的矩陣元素，並且我們可以對這個 矩陣進行對角化，以找到它的特徵值和特徵態。這個練習顯示了基態能量是
and that the corresponding wavefunction is 並且對應的波函數是
where is a normalization constant. To the extent that involves several Hartree-Fock type wavefunctions, and the corresponding energy is lower than all other approximations we tried, this represents the optimal solution to the problem within our choice of basis, including correlation effects. A study of the ground state as a function of the parameter elucidates the effects of correlation between the electrons in this simple model (see Problem 5). A more accurate description should include the excited states of electrons in each atom, which increases significantly the size of the matrices involved. Extending this picture to more complex systems produces an almost exponential increase of the computational difficulty. 其中 是一個正規化常數。在 涉及幾個 Hartree-Fock 類型的波函數，並且對應的能量低於我們嘗試的所有其他近似值時，這代表了在我們選擇的基礎上解決問題的最佳解決方案，包括相關效應。對參數 的地面狀態的研究闡明了在這個簡單模型中電子之間的相關效應（見問題 5）。更準確的描述應該包括每個原子中電子的激發態，這會顯著增加所涉及矩陣的大小。將這一圖像擴展到更複雜的系統會導致計算困難度的幾乎指數級增加。
2.5 Density Functional Theory 2.5 密度泛函理論
In a series of seminal papers, Hohenberg, Kohn and Sham developed a different way of looking at the problem, which has been called Density Functional 在一系列開創性的論文中，Hohenberg、Kohn 和 Sham 提出了一種不同的觀點來看待這個問題，這被稱為密度泛函
Theory (DFT). The basic ideas of Density Functional Theory are contained in the two original papers of Hohenberg, Kohn and Sham, [22, 23] and are referred to as the Hohenberg-Kohn-Sham theorem. This theory has had a tremendous impact on realistic calculations of the properties of molecules and solids, and its applications to different problems continue to expand. A measure of its importance and success is that its main developer, W. Kohn (a theoretical physicist) shared the 1998 Nobel prize for Chemistry with J.A. Pople (a computational chemist). We will review here the essential ideas behind Density Functional Theory. 密度泛函理論（DFT）。密度泛函理論的基本思想包含在 Hohenberg、Kohn 和 Sham 的兩篇原始論文中，[22, 23]，並被稱為 Hohenberg-Kohn-Sham 定理。這個理論對分子和固體性質的實際計算產生了巨大影響，其應用於不同問題的範圍不斷擴大。其重要性和成功的一個衡量標準是，其主要開發者 W. Kohn（一位理論物理學家）與 J.A. Pople（一位計算化學家）共同獲得了 1998 年的諾貝爾化學獎。我們將在這裡回顧密度泛函理論背後的基本思想。
The basic concept is that instead of dealing with the many-body Schrödinger equation, Eq. (2.1), which involves the many-body wavefunction , one deals with a formulation of the problem that involves the total density of electrons . This is a huge simplification, since the many-body wavefunction need never be explicitly specified, as was done in the Hartree and Hartree-Fock approximations. Thus, instead of starting with a drastic approximation for the behavior of the system (which is what the Hartree and Hartree-Fock wavefunctions represent), one can develop the appropriate single-particle equations in an exact manner, and then introduce approximations as needed. 基本概念是，與涉及多體波函數 的多體 Schrödinger 方程式（2.1 式）不同，人們處理的是涉及電子總密度 的問題陳述。這是一個巨大的簡化，因為多體波函數永遠不需要明確指定，就像在 Hartree 和 Hartree-Fock 近似中所做的那樣。因此，與其從系統行為的嚴重近似（這就是 Hartree 和 Hartree-Fock 波函數所代表的）開始，人們可以以精確的方式發展適當的單粒子方程式，然後根據需要引入近似。
In the following discussion we will make use of the density and the oneparticle and two-particle density matrices, denoted by , respectively, as expressed through the many-body wavefunction: 在以下討論中，我們將使用密度 以及單粒子和雙粒子密度矩陣，分別表示為 ，如通過多體波函數表達的那樣：
These quantities are defined in detail in Appendix B, Eqs. (B.13)-(B.15), where their physical meaning is also discussed. 這些數量在附錄 B 中詳細定義，方程式(B.13)-(B.15)中也討論了它們的物理意義。
First, we will show that the density is uniquely defined given an external potential for the electrons (this of course is identified with the ionic potential). To prove this, suppose that two different external potentials, and , give rise to the same density . We will show that this is impossible. We assume that and are different in a non-trivial way, that is, they do not differ merely by a constant. Let and be the total energy and wavefunction and and be the total energy and wavefunction for the systems with hamiltonians and , respectively, where the first hamiltonian contains and the second as external potential: 首先，我們將展示對於電子的外部電位 ，密度 是唯一確定的（這當然與離子電位相對應）。為了證明這一點，假設兩個不同的外部電位 和 導致相同的密度 。我們將展示這是不可能的。我們假設 和 以一種非平凡的方式不同，即它們不僅僅是一個常數的差異。讓 和 分別是具有哈密頓量 和 的系統的總能量和波函數，其中第一個哈密頓量包含 ，第二個包含 作為外部電位：
Then we will have, by the variational principle, 然後我們將通過變分原理得到，
where the strict inequality is a consequence of the fact that the two potentials are different in a non-trivial way. Similarly we can prove 在這裡，嚴格不等式是兩個電位以非平凡方式不同的結果。同樣地，我們可以證明
Adding Eqs. (2.71) and (2.72), we obtain 將方程式（2.71）和（2.72）相加，我們得到
But the last two terms on the right-hand side of Eq. (2.73) give 但是方程（2.73）右側的最後兩項給。
because by assumption the densities and corresponding to the two potentials are the same. This leads to the relation , which is obviously wrong; therefore we conclude that our assumption about the densities being the same cannot be correct. This proves that there is a one-to-one correspondence between an external potential and the density . But the external potential determines the wavefunction, so that the wavefunction must be a unique functional of the density. If we denote as the terms in the hamiltonian other than , with representing the kinetic energy and the electron-electron interaction, we conclude that the expression 因為假設對應於兩個電位的密度 和 是相同的。這導致關係 ，顯然是錯誤的；因此我們得出結論，我們關於密度相同的假設是不正確的。這證明了外部電位 和密度 之間存在一對一的對應關係。但外部電位確定了波函數，因此波函數必須是密度的唯一函數。如果我們將哈密頓量中除 以外的項表示為 ，其中 代表動能， 代表電子間相互作用，我們得出以下表達式
must be a universal functional of the density, since the terms and , the kinetic energy and electron-electron interactions, are common to all solids, and therefore this functional does not depend on anything else other than the electron density (which is determined uniquely by the external potential that differs from system to system). 必須是密度的通用功能，因為動能和電子-電子相互作用這兩個項目對所有固體都是共通的，因此這個功能不依賴於除了電子密度之外的任何其他東西（這是由外部電位 唯一確定的，該電位因系統而異）。
From these considerations we conclude that the total energy of the system is a functional of the density, and is given by 從這些考慮中，我們得出結論，系統的總能量是密度的一個泛函，並且由此給出
From the variational principle we can deduce that this functional attains its minimum for the correct density corresponding to , since for a given and any 從變分原理我們可以推斷出，這個泛函在對應於 的正確密度 時達到最小值，因為對於給定的 和任何
other density we would have 其他密度 我們會有
Using our earlier expressions for the one-particle and the two-particle density matrices, we can obtain explicit expressions for and : 使用我們先前對單粒子和雙粒子密度矩陣的表達式，我們可以得到 和 的明確表達式：
Now we can attempt to reduce these expressions to a set of single-particle equations, as before. The important difference in the present case is that we do not have to interpret these single-particle states as corresponding to electrons. They represent fictitious fermionic particles with the only requirement that their density is identical to the density of the real electrons. These particles can be considered to be non-interacting: this is a very important aspect of the nature of the fictitious particles, which will allow us to simplify things considerably, since their behavior will not be complicated by interactions. The assumption that we are dealing with non-interacting particles can be exploited to express the many-body wavefunction in the form of a Slater determinant, as in Eq. (2.14). We can then express the various physical quantities in terms of the single-particle orbitals that appear in the Slater determinant. We obtain 現在我們可以嘗試將這些表達式簡化為一組單粒子方程式，就像以前一樣。目前情況的重要差異在於，我們不必將這些單粒子狀態解釋為對應到電子。它們代表虛構的費米子粒子，唯一的要求是它們的密度與真實電子的密度相同。這些粒子可以被認為是非相互作用的：這是虛構粒子性質的一個非常重要的方面，這將使我們能夠大大簡化事情，因為它們的行為不會被相互作用所複雜化。我們正在處理非相互作用粒子的假設可以被利用來以斯萊特行列式的形式表達多體波函數 ，如(2.14)式所示。然後我們可以用出現在斯萊特行列式中的單粒子軌道 來表達各種物理量。我們得到
With the help of Eqs. (2.79)-(2.81), we can express the various terms in the energy functional, which take the form 借助 Eqs. (2.79)-(2.81)的幫助，我們可以表達能量泛函中的各種項，其形式為
In this expression, the first term represents the kinetic energy of the states in the Slater determinant (hence the superscript ). Since the fictitious particles are noninteracting, we can take the kinetic energy to be given by 在這個表達式中，第一項代表了 Slater 行列式中狀態的動能（因此上標 ）。由於虛構粒子是非相互作用的，我們可以將動能表示為
This is a conveniently simple model for illustrating electron exchange and correlation effects. It is discussed in several of the textbooks mentioned in chapter 1. 這是一個方便簡單的模型，用於說明電子交換和相關效應。它在第 1 章提到的幾本教科書中有討論。
