**2.1 Quantum mechanics in a nutshell**

For the benefit of those readers who are unfamiliar with the standard formalism of quantum mechanics, we review its main facts:


#### **2.2 Naive tight-binding approach to band theory**

As argued in Section 1, the starting point of this approach is to express the electron's state vector as a linear combination of atomic orbitals (LCAO) located at the crystal's lattice sites (we illustrate the procedure in 1D, but clearly, it can be extended to any dimension and lattice symmetry). The eigenvalue problem of the isolated atom centered at *xc* is *H*atom ∣*ψ*atom⟩ ¼ *E*atom ∣*ψ*atom⟩, with *H*atom ¼ *T* þ *V x*ð Þ � *xc* . We then place a copy ∣*i*⟩ of ∣*ψatom*⟩ centered at each lattice site *i* (*xi* ¼ *ia*) and write the electron's state in the crystal as LCAO

$$|\psi\_{\text{crystal}}\rangle = \sum\_{i} c\_{i} |i\rangle \tag{1}$$

*Assuming* the states ∣*i*⟩ to be *orthogonal* to each other, the left-hand side of Eq. (3) reads ∑*<sup>i</sup> ci*∣*i*i � *γ* ∑*<sup>i</sup> ci* ð j*i* þ 1i þ *ci*þ<sup>1</sup>∣*i*⟩). If the number of sites in the crystal is large

Again invoking PBC, one tries the form *cj* ¼ exp *ijka* with �*π* , *ka*≤*π* (Bloch

• We know that *α* equals *Eatom* plus some correction, but we do not know what

• Similarly, we know that *γ* is the expectation value of the effective potential

atoms. We have kept just *j* ¼ *i* � 1, but even in this approximation, we do not

• To what extent can one assume the states ∣*i*⟩ to be *orthogonal* to each other? This assumption is correct in the absence of interatomic interaction, but not

*<sup>i</sup>* þ *Wi*

periodic boundary conditions (PBC). This allows to rearrange the sums (their indices become dummy), and Eq. (3) reads <sup>∑</sup>*<sup>i</sup> <sup>E</sup>*crystal � *<sup>α</sup>* � �*ci* � *<sup>γ</sup>* <sup>∑</sup>*<sup>i</sup>* <sup>ð</sup>*ci*�1<sup>þ</sup> � *ci*þ1Þ�∣*i*i ¼ 0 . Clearly, the LCAO assumes that the ∣*i*⟩ are *linearly independent* (be they orthogonal or not), so we are left with the system of difference equations:

*i*

*Ecrystal* � *<sup>α</sup>* � �*ci* � *<sup>γ</sup>* <sup>∑</sup>

phase factors) and obtains the known cosine spectrum

What has been left behind? Much indeed:

), one can greatly simplify the problem by assuming

*E*crystal ¼ *α* � 2*γ* cos *ka,* � *π* , *ka* ≤*π:* (5)

� � felt by an electron at *<sup>x</sup>* � *ia*, due to the presence of other

� � and using Eq. (1), *E*crystal turns out

*ci* j j<sup>2</sup> <sup>þ</sup> <sup>∑</sup> *ij Sijc* <sup>∗</sup> *<sup>i</sup> cj*

" #

*,* (6)

� � can be singled out from

involve only sites *i* and *j*,

∑ *i*

*<sup>α</sup><sup>i</sup>* <sup>≔</sup> *Hii* <sup>¼</sup> 〈*i*∣*Wi*∣*i*〉*, <sup>γ</sup>ij* <sup>≔</sup> *Hij* <sup>¼</sup> *<sup>i</sup>*j*Wi*<sup>j</sup> *<sup>j</sup>*i*, Sij* <sup>≔</sup> <sup>h</sup>*i*<sup>j</sup> *<sup>j</sup>*i*, j* 6¼ *<sup>i</sup>:* � (7)

The contribution of the *Sij* (known as *overlap integrals*) to the band spectrum is our main concern in this section. But not less interesting are that of the *α<sup>i</sup>* terms which, as argued, shift the electronic energy in an atom from its isolated value *E*atom, as a collective effect of the other atoms—and that of the *γij*. The latter can be

ð Þ2

*ij* <sup>≔</sup> *<sup>i</sup>*j*Vj*<sup>j</sup> *<sup>j</sup>* � �

*ij* also involve the sum <sup>∑</sup>*<sup>l</sup>*6¼*i,j V x*ð Þ � *xl* of the potentials

ð Þ¼ *ci*�<sup>1</sup> þ *ci*þ<sup>1</sup> 0*, i* ¼ 1…*N* � 0*:* (4)

enough (usually it is �10<sup>6</sup>

*Issues in Solid-State Physics*

*DOI: http://dx.doi.org/10.5772/intechopen.84367*

the correction is.

*Wi* ≔ ∑*<sup>j</sup>*6¼*<sup>i</sup> V x* � *xj*

to be [1, 2]

where

**7**

know what the correction is.

necessarily when atoms interact.

Recognizing that *<sup>H</sup>*crystal <sup>¼</sup> <sup>∑</sup>*<sup>i</sup> <sup>H</sup>*atom

*E*crystal ¼ *E*atom þ ∑

**2.3 Tight-binding band calculation: properly done**

*i*

regarded as the sum of two contributions, as *Vj* ≔ *V x* � *xj*

ð Þ3

*Wi*. Then whereas the two-center integrals *γ*

the three-center integrals *γ*

*<sup>α</sup><sup>i</sup> ci* j j<sup>2</sup> <sup>þ</sup> <sup>∑</sup>

*ij γijc* <sup>∗</sup> *<sup>i</sup> cj*

" #�

(clearly, *ci* ¼ 〈*i*∣*ψ*crystal〉). Now, even though the interatomic distance in the crystal (the "lattice spacing" *a*) is usually larger than the range *x*<sup>0</sup> of the atomic orbitals, the atomic cores *do* interact, and one should include at least two effects:


It thus makes sense to write up the lattice Hamiltonian in terms of projection operators as

$$H\_{\text{crystal}} = \alpha \sum\_{i} |i\rangle\langle i| - \gamma \sum\_{i} (|i+\mathbf{1}\rangle\langle i| + |i\rangle\langle i+\mathbf{1}|). \tag{2}$$

Two timely comments are:


Using Eqs. (1) and (2), the eigenvalue problem *H*crystal ∣*ψ*crystal⟩ ¼ *E*crystal ∣*ψ*crystal⟩ for the electron in the crystal reads

$$
\nabla \left[ a \sum\_{i} |i\rangle \langle i| - \chi \sum\_{i} (|i+1\rangle \langle i| + |i\rangle \langle i+1|) \right] \sum\_{j} c\_{j} |j\rangle = E\_{\text{crystal}} \sum\_{j} c\_{j} |j\rangle. \tag{3}
$$

#### *Issues in Solid-State Physics DOI: http://dx.doi.org/10.5772/intechopen.84367*

• **Wave function**: a possible "basis" set is that of eigenstates (*X*∣*x*⟩ ¼ *x*∣*x*⟩) of the

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

here the role of the coefficients *αI*. In modern notation, *ψ*ð Þ *x* is written as h i *x*j*φ* ,

• **Orthogonality** (h i *φ*j*ψ* ¼ 0): a given eigenvalue *l*<sup>1</sup> of a Hermitian operator *L* may have a single eigenstate ∣ *l*1⟩ (the normalized one out of a dimension-1 subspace over the complex numbers) or more (in this case, it is said to be *degenerate*). Eigenstates corresponding to *different* eigenvalues are *automatically*

As argued in Section 1, the starting point of this approach is to express the electron's state vector as a linear combination of atomic orbitals (LCAO) located at the crystal's lattice sites (we illustrate the procedure in 1D, but clearly, it can be extended to any dimension and lattice symmetry). The eigenvalue problem of the

*H*atom ¼ *T* þ *V x*ð Þ � *xc* . We then place a copy ∣*i*⟩ of ∣*ψatom*⟩ centered at each lattice

∣*ψ*crystal〉 ¼ ∑

(clearly, *ci* ¼ 〈*i*∣*ψ*crystal〉). Now, even though the interatomic distance in the crystal (the "lattice spacing" *a*) is usually larger than the range *x*<sup>0</sup> of the atomic orbitals, the atomic cores *do* interact, and one should include at least two effects:

• A *correction* to the isolated atomic level *E*atom (we shall call *α* the corrected

• Electron tunneling between neighboring orbitals (let *γ* be a gauge of the energy

It thus makes sense to write up the lattice Hamiltonian in terms of projection

*i*

2. The minus sign in the second term ensures crystal stability (energy is released

Using Eqs. (1) and (2), the eigenvalue problem *H*crystal ∣*ψ*crystal⟩ ¼ *E*crystal ∣*ψ*crystal⟩

*j*

*cj*∣ *j*i ¼ *E*crystal ∑

*j*

*cj*∣ *j*i*:* (3)

j*i*i〈*i*∣ � *γ* ∑

1. The presence of ∣*i*⟩⟨*i* þ 1∣, the adjoint of ∣*i* þ 1⟩⟨*i*∣, ensures that *H*crystal be

ðj*i* þ 1i〈*i*∣þj*i*i〈*i* þ 1∣Þ� ∑

*i*

*dx ψ*ð Þ *x* ∣*x*⟩. The *wave function ψ*ð Þ *x* plays

*ci*∣*i*i (1)

ðj*i* þ 1ih*i*jþj*i*ih Þ *i* þ 1j *:* (2)

position operator, namely, <sup>∣</sup>*ψ*⟩ <sup>¼</sup> <sup>Ð</sup>

**2.2 Naive tight-binding approach to band theory**

involved in such a "hopping" process)

*H*crystal ¼ *α* ∑

Two timely comments are:

by forming a crystal).

for the electron in the crystal reads

j*i*i〈*i*∣ � *γ* ∑

*i*

½*α* ∑ *i*

**6**

*Hermitian*.

*i*

*dx* ∣*x*⟩h i *x*j*φ* .

isolated atom centered at *xc* is *H*atom ∣*ψ*atom⟩ ¼ *E*atom ∣*ψ*atom⟩, with

site *i* (*xi* ¼ *ia*) and write the electron's state in the crystal as LCAO

so one writes <sup>∣</sup>*φ*⟩ <sup>¼</sup> <sup>Ð</sup>

orthogonal.

level)

operators as

*Assuming* the states ∣*i*⟩ to be *orthogonal* to each other, the left-hand side of Eq. (3) reads ∑*<sup>i</sup> ci*∣*i*i � *γ* ∑*<sup>i</sup> ci* ð j*i* þ 1i þ *ci*þ<sup>1</sup>∣*i*⟩). If the number of sites in the crystal is large enough (usually it is �10<sup>6</sup> ), one can greatly simplify the problem by assuming periodic boundary conditions (PBC). This allows to rearrange the sums (their indices become dummy), and Eq. (3) reads <sup>∑</sup>*<sup>i</sup> <sup>E</sup>*crystal � *<sup>α</sup>* � �*ci* � *<sup>γ</sup>* <sup>∑</sup>*<sup>i</sup>* <sup>ð</sup>*ci*�1<sup>þ</sup> � *ci*þ1Þ�∣*i*i ¼ 0 . Clearly, the LCAO assumes that the ∣*i*⟩ are *linearly independent* (be they orthogonal or not), so we are left with the system of difference equations:

$$(\mathbf{E}\_{\text{crystal}} - \mathbf{a})\mathbf{c}\_i - \chi \sum\_{i} (\mathbf{c}\_{i-1} + \mathbf{c}\_{i+1}) = \mathbf{0}, i = \mathbf{1}...N \equiv \mathbf{0}.\tag{4}$$

Again invoking PBC, one tries the form *cj* ¼ exp *ijka* with �*π* , *ka*≤*π* (Bloch phase factors) and obtains the known cosine spectrum

$$E\_{\text{crystal}} = a - 2\gamma \cos ka, \ -\pi \le ka \le \pi. \tag{5}$$

What has been left behind? Much indeed:


#### **2.3 Tight-binding band calculation: properly done**

Recognizing that *<sup>H</sup>*crystal <sup>¼</sup> <sup>∑</sup>*<sup>i</sup> <sup>H</sup>*atom *<sup>i</sup>* þ *Wi* � � and using Eq. (1), *E*crystal turns out to be [1, 2]

$$E\_{\text{crystal}} = E\_{\text{atom}} + \left[\sum\_{i} a\_i |c\_i|^2 + \sum\_{ij} \gamma\_{ji} c\_i^\* c\_j\right] \Big/ \left[\sum\_i |c\_i|^2 + \sum\_{ij} \mathbf{S}\_{ji} c\_i^\* \, c\_j\right],\tag{6}$$

where

$$a\_i \coloneqq H\_{\vec{n}} = \langle i | W\_i | i \rangle, \gamma\_{\vec{\eta}} \coloneqq H\_{\vec{\eta}} = \langle i | \mathcal{W}\_i | j \rangle, \mathcal{S}\_{\vec{\eta}} \coloneqq \langle i | j \rangle, j \neq i. \tag{7}$$

The contribution of the *Sij* (known as *overlap integrals*) to the band spectrum is our main concern in this section. But not less interesting are that of the *α<sup>i</sup>* terms which, as argued, shift the electronic energy in an atom from its isolated value *E*atom, as a collective effect of the other atoms—and that of the *γij*. The latter can be regarded as the sum of two contributions, as *Vj* ≔ *V x* � *xj* � � can be singled out from *Wi*. Then whereas the two-center integrals *γ* ð Þ2 *ij* <sup>≔</sup> *<sup>i</sup>*j*Vj*<sup>j</sup> *<sup>j</sup>* � � involve only sites *i* and *j*, the three-center integrals *γ* ð Þ3 *ij* also involve the sum <sup>∑</sup>*<sup>l</sup>*6¼*i,j V x*ð Þ � *xl* of the potentials of the remaining atoms in the solid. Hence, the *γ* ð Þ3 *ij* can be interpreted as the collective effect on the overlap between orbitals *i* and *j*.

Variation of Eq. (6) with respect to the LCAO coefficients of Eq. (1)—namely, *∂E*crystal*=∂a*<sup>∗</sup> *<sup>j</sup>* <sup>¼</sup> <sup>0</sup>—yields *Hij* � *<sup>E</sup>*crystal*Sij aj* <sup>¼</sup> <sup>0</sup>*,* <sup>∀</sup>*j*. Assuming PBC, *Hij* and *Sij* are functions only of the interatomic distance *na*, with *n* ¼ ∣*i* � *j*∣. Again using *an* ¼ exp *inka* with �*π* , *ka*≤ *π*, Eq. (6) yields

$$E\_{\text{crystal}} = E\_{\text{atom}} + \left[a + 2\sum\_{n} H\_n \cos nka\right] \Big/ \left[\mathbf{1} + 2\sum\_{n} \mathbf{S}\_n \cos nka\right]. \tag{8}$$

**3. Quantum Monte Carlo method for systems with strongly**

The *state vectors* dealt with in Section 1 represent *pure* states. They are the ones

*bles* of quantum states, the object of interest is the Hermitian operator exp ð Þ �*βH* , called the *density matrix* operator (here <sup>β</sup> <sup>≔</sup> ð Þ *kBT* �<sup>1</sup> and *kB* <sup>¼</sup> *<sup>R</sup>=NA* are Boltzmann'<sup>s</sup>

What drives our interest in the *density matrix*—namely, the matrix elements between pure states of exp ð Þ �*βH* —is the fact that it can be used to find the ground state of many-body systems by stochastic methods. For *β* large enough, exp ð Þ �*βH* acts effectively as a *projector* over the lowest-lying energy eigenstate to which the initial (trial) state ∣*φ*⟩ is not definitely orthogonal. Let *E* be the corresponding eigenvalue, and consider another trial state ∣*χ*⟩ over which we will project the result.

But what is yet more interesting is that in the process, we find a good estimate of

The first step in this computation is to divide the interval 0½ � *; β* into *L* "time"

a.We take our language from the formal analogy between the density matrix

b.Note that in our case, exp ð Þ �*βH* is *not meant to be traced over* as it should be in a thermodynamic calculation: here it must rather be considered as a formal

If we can decompose *H* into a sum of several terms *Hi* which (although not commuting among them) are themselves *sums of commuting terms*, then for *L* large

*<sup>U</sup>* <sup>¼</sup> exp ½�Δ*τ*ð Þ *<sup>H</sup>*<sup>1</sup> <sup>þ</sup> <sup>2</sup> � ¼ exp ð Þ �Δ*τ*<sup>1</sup> exp ð Þ �Δ*τH*<sup>2</sup> exp �ð Þ <sup>Δ</sup>*<sup>τ</sup>* <sup>2</sup>

. Hence

<sup>3</sup> We have already stated that the Fermi level is the *chemical potential* of an ideal free-electron gas. This

the eigenstate itself, namely, its composition in terms of a known basis.

*<sup>β</sup>*!<sup>∞</sup> ½ � ⟨ *<sup>χ</sup>*<sup>j</sup> exp ½�ð Þ *<sup>β</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>β</sup> <sup>H</sup>*�j*φ*⟩*=*⟨ *<sup>χ</sup>*<sup>j</sup> exp ð Þj �Δ*β<sup>H</sup> <sup>φ</sup>*⟩ *:* (11)

<sup>3</sup> When one deals with *statistical ensem-*

½ � *H*1*; H*<sup>2</sup> n o

which display the spectacular effects seen in recent experiments. Since in this section, we will allow *creation annihilation* of electron states, we must work in the

**correlated fermions**

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constant).

**3.1 Quantum** *statistical* **mechanics in a nutshell**

framework of the *grand canonical ensemble*.

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Then we may numerically compute *E* from

**3.2 Monte Carlo pursuit of the ground state**

tool to make sense in the limit *β* ! ∞.

c. We may call *U* ¼ exp ð Þ �Δ*τH* the *transfer matrix* operator.

½ �þ *H*1*; H*<sup>2</sup> … n o � *<sup>U</sup>*1*U*<sup>2</sup>

and the evolution operators.

enough, the error of approximating

<sup>¼</sup> *<sup>U</sup>*1*U*<sup>2</sup> <sup>1</sup> � ð Þ <sup>Δ</sup>*<sup>τ</sup>* <sup>2</sup>

**9**

would be at most of order ð Þ <sup>Δ</sup>*<sup>τ</sup>* <sup>2</sup>

concept is peculiar of the *grand canonical ensemble*.

slices of width Δ*τ* ¼ *β=L*. Some comments are in order:

exp ð Þ¼ �Δ*βE* lim

Note however that the number of multicenter integrals to be computed is immense! Because of that, most tight-binding calculations plainly ignore almost all the multicenter integrals (keeping only those involving nearest neighbors) and neglect orbital non-orthogonality. This way, the familiar cosine spectrum is obtained. Often, multicenter integrals are just regarded as parameters to fit the results of more sophisticated calculations made by other methods at the highest symmetry points of the Brillouin zone.

In the following, we compute *all* the multicenter integrals *exactly* in the framework of a simple model for the atomic potential. The results help get an intuition on the effect on band spectrum of neglecting overlap integrals and distant-neighbor interactions.

#### **2.4 A simple model that yields an exact tight-binding band spectrum**

We restrict ourselves to a 1D monoatomic crystal and assume the interatomic distance *a* to be larger than the effective range of the screened Coulomb potential representing the atomic core. In such a situation, we can approximate the latter by a Dirac *δ*-function (complete screening up to the scale of the nucleus):

$$V\_{\text{crystal}}(\mathbf{x}) = -V\_0 \sum\_n \delta(\mathbf{x} - na). \tag{9}$$

The solution to *<sup>H</sup>*atom <sup>∣</sup>*ψ*atom⟩ <sup>¼</sup> *<sup>E</sup>*atom <sup>∣</sup>*ψ*atom⟩, with *<sup>H</sup>*atom ¼ � <sup>ℏ</sup><sup>2</sup> 2*m d*2 *dx*<sup>2</sup> � *<sup>V</sup>*0*δ*ð Þ *<sup>x</sup>* , is an exponential function of the form *ψ*atomð Þ¼ *x* 〈*x*j i *ψ*atom ¼ *x* �1 2 <sup>0</sup> exp � j j *<sup>x</sup> x*0 . Its range *<sup>x</sup>*<sup>0</sup> is related to *<sup>E</sup>*atom by�*E*atom <sup>¼</sup> <sup>ℏ</sup><sup>2</sup> *=*2*mx*<sup>2</sup> <sup>0</sup> <sup>¼</sup> *mV*0*=*2ℏ<sup>2</sup> .

The only two spatial scales involved in this problem are *x*<sup>0</sup> and the lattice spacing *a*. The parameter *t* ¼ *a=x*<sup>0</sup> will thus allow us to follow the formation of energy bands (*k*-space picture) as atoms get close together (real-space picture). All the multicenter integrals can be computed analytically in terms of *t*. The results are *Sn* ¼ ð Þ 1 þ *nt* exp ð Þ �*t* , *α* ¼ 2*E*atom exp ð Þ �*t =*sinh *t*, and *γ<sup>n</sup>* ¼ 2*E*atom½ � *n* þ exp ð Þ �*t =*sinh *t* exp ð Þ �*nt* [3]. We thus get the following closed expression for *<sup>λ</sup>* <sup>≔</sup> *<sup>E</sup>*crystal � *<sup>E</sup>*atom *<sup>=</sup>E*atom:

$$\lambda(k,t) = \left[A\_0(t) + A\_1(t)\cos ka\right] / \left[1 + \mathcal{S}(t)\cos ka\right], \ -\pi \le ka \le \pi,\tag{10}$$

where *A*<sup>0</sup> ¼ exp ð Þ �*t* sinh *t=*½ � sinh *t*cosh *t* � *t* , *A*<sup>1</sup> ¼ sinh *t=*½ � sinh *t*cosh *t* � *t* , and *S* ¼ ½ � *t*cosh *t* � sinh *t =*½ � sinh *t*cosh *t* � *t* [3].

Explicit evaluation of Eq. (10) at the bottom (*ka* ¼ 0) and top (*ka* ¼ *π*) of the band shows that for *t* , 4, the cosine spectrum of Eq. (5) underestimates both. Moreover, the multicenter integrals neglected in the cosine spectrum shift *unevenly* the top and bottom of the exact spectrum. Hence, the approximation performs worse for the top than for the bottom of the band.

of the remaining atoms in the solid. Hence, the *γ*

*<sup>j</sup>* ¼ 0—yields *Hij* � *E*crystal*Sij*

*an* ¼ exp *inka* with �*π* , *ka*≤ *π*, Eq. (6) yields

*E*crystal ¼ *E*atom þ *α* þ 2 ∑

symmetry points of the Brillouin zone.

*∂E*crystal*=∂a*<sup>∗</sup>

interactions.

collective effect on the overlap between orbitals *i* and *j*.

ð Þ3

*aj* <sup>¼</sup> <sup>0</sup>*,* <sup>∀</sup>*j*. Assuming PBC, *Hij* and *Sij* are

1 þ 2 ∑ *n*

Variation of Eq. (6) with respect to the LCAO coefficients of Eq. (1)—namely,

*Hn* cos *nka* 

In the following, we compute *all* the multicenter integrals *exactly* in the framework of a simple model for the atomic potential. The results help get an intuition on the effect on band spectrum of neglecting overlap integrals and distant-neighbor

We restrict ourselves to a 1D monoatomic crystal and assume the interatomic distance *a* to be larger than the effective range of the screened Coulomb potential representing the atomic core. In such a situation, we can approximate the latter by a

*=*2*mx*<sup>2</sup>

*λ*ð Þ¼ *k; t* ½ � *A*0ðÞþ*t A*1ð Þ*t* cos *ka =*½ � 1 þ *S t*ð Þ cos *ka ,* � *π* , *ka*≤ *π,* (10)

where *A*<sup>0</sup> ¼ exp ð Þ �*t* sinh *t=*½ � sinh *t*cosh *t* � *t* , *A*<sup>1</sup> ¼ sinh *t=*½ � sinh *t*cosh *t* � *t* , and

Explicit evaluation of Eq. (10) at the bottom (*ka* ¼ 0) and top (*ka* ¼ *π*) of the band shows that for *t* , 4, the cosine spectrum of Eq. (5) underestimates both. Moreover, the multicenter integrals neglected in the cosine spectrum shift *unevenly* the top and bottom of the exact spectrum. Hence, the approximation performs

The only two spatial scales involved in this problem are *x*<sup>0</sup> and the lattice spacing

*a*. The parameter *t* ¼ *a=x*<sup>0</sup> will thus allow us to follow the formation of energy bands (*k*-space picture) as atoms get close together (real-space picture). All the multicenter integrals can be computed analytically in terms of *t*. The results are

*γ<sup>n</sup>* ¼ 2*E*atom½ � *n* þ exp ð Þ �*t =*sinh *t* exp ð Þ �*nt* [3]. We thus get the following closed

*V*crystalð Þ¼� *x V*0∑*nδ*ð Þ *x* � *na :* (9)

<sup>0</sup> <sup>¼</sup> *mV*0*=*2ℏ<sup>2</sup> .

2*m d*2

�1 2 <sup>0</sup> exp � j j *<sup>x</sup>*

*dx*<sup>2</sup> � *<sup>V</sup>*0*δ*ð Þ *<sup>x</sup>* , is

. Its

*x*0 

Note however that the number of multicenter integrals to be computed is immense! Because of that, most tight-binding calculations plainly ignore almost all the multicenter integrals (keeping only those involving nearest neighbors) and neglect orbital non-orthogonality. This way, the familiar cosine spectrum is obtained. Often, multicenter integrals are just regarded as parameters to fit the results of more sophisticated calculations made by other methods at the highest

functions only of the interatomic distance *na*, with *n* ¼ ∣*i* � *j*∣. Again using

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

*n*

**2.4 A simple model that yields an exact tight-binding band spectrum**

Dirac *δ*-function (complete screening up to the scale of the nucleus):

The solution to *<sup>H</sup>*atom <sup>∣</sup>*ψ*atom⟩ <sup>¼</sup> *<sup>E</sup>*atom <sup>∣</sup>*ψ*atom⟩, with *<sup>H</sup>*atom ¼ � <sup>ℏ</sup><sup>2</sup>

an exponential function of the form *ψ*atomð Þ¼ *x* 〈*x*j i *ψ*atom ¼ *x*

*Sn* ¼ ð Þ 1 þ *nt* exp ð Þ �*t* , *α* ¼ 2*E*atom exp ð Þ �*t =*sinh *t*, and

*=E*atom:

range *<sup>x</sup>*<sup>0</sup> is related to *<sup>E</sup>*atom by�*E*atom <sup>¼</sup> <sup>ℏ</sup><sup>2</sup>

expression for *λ* ≔ *E*crystal � *E*atom

**8**

*S* ¼ ½ � *t*cosh *t* � sinh *t =*½ � sinh *t*cosh *t* � *t* [3].

worse for the top than for the bottom of the band.

*ij* can be interpreted as the

*Sn* cos *nka* 

*:* (8)
