Applications of Density Functional Theory (DFT) in Various Fields

WHAT IS DENSITY FUNCTIONAL

WHAT IS DENSITY FUNCTIONAL

THEORY?

THEORY?

1.1 EXAMPLES OF DFT IN ACTION

1.1 EXAMPLES OF DFT IN ACTION

•

1.1.1 

1.1.1 

Ammonia Synthesis by Heterogeneous Catalysis

Ammonia Synthesis by Heterogeneous Catalysis

 The core reaction in ammonia production is very simple:

 The core reaction in ammonia production is very simple:

N

N

2

2

 +3H

 +3H

2

2

→

→

 2NH

 2NH

3

3

Can a direct connection be made between the shape and

Can a direct connection be made between the shape and

size of a metal nanoparticle and its activity as a catalyst for

size of a metal nanoparticle and its activity as a catalyst for

ammonia synthesis?

ammonia synthesis?

2

Honkala and co-workers, using DFT calculations, showed that the net

Honkala and co-workers, using DFT calculations, showed that the net

chemical reaction above proceeds via at least 12 distinct steps on a

chemical reaction above proceeds via at least 12 distinct steps on a

metal catalyst and that the rates of these steps depend strongly on

metal catalyst and that the rates of these steps depend strongly on

the local coordination of the metal atoms that are involved.

the local coordination of the metal atoms that are involved.

O

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3

1.1.2 

1.1.2 

Embrittlement of Metals by Trace Impurities

Embrittlement of Metals by Trace Impurities

It has been known for over 100 years that adding tiny amounts of

It has been known for over 100 years that adding tiny amounts of

certain impurities to copper can change the metal from being ductile

certain impurities to copper can change the metal from being ductile

to a material that will fracture in a brittle way. This occurs, for

to a material that will fracture in a brittle way. This occurs, for

example, when bismuth (Bi) is present in copper(Cu) at levels below

example, when bismuth (Bi) is present in copper(Cu) at levels below

100 ppm.

100 ppm.

Can the changes in copper caused by Bi be explained in a

Can the changes in copper caused by Bi be explained in a

detailed way?

detailed way?

4

Schweinfest, Paxton, and Finnis used DFT calculations to

Schweinfest, Paxton, and Finnis used DFT calculations to

offer a definitive description of how Bi embrittles copper.

offer a definitive description of how Bi embrittles copper.

They first used DFT to 

They first used DFT to 

predict stress–strain relationships

predict stress–strain relationships

for pure Cu and Cu containing Bi atoms as impurities

for pure Cu and Cu containing Bi atoms as impurities

. They

. They

explicitly calculated 

explicitly calculated 

the cohesion energy of a particular

the cohesion energy of a particular

grain boundary 

grain boundary 

that is known experimentally to be

that is known experimentally to be

embrittled by Bi.

embrittled by Bi.

5

1.1.3 

1.1.3 

Materials Properties for Modeling Planetary Formation

Materials Properties for Modeling Planetary Formation

The extreme conditions that exist inside planets pose some

The extreme conditions that exist inside planets pose some

obvious challenges to probing these topics in laboratory

obvious challenges to probing these topics in laboratory

experiments.

experiments.

DFT calculations can play a useful role in probing material

DFT calculations can play a useful role in probing material

properties at these extreme conditions.

properties at these extreme conditions.

6

Umemoto et al. wanted to understand what happens to the

Umemoto et al. wanted to understand what happens to the

structure of MgSiO

structure of MgSiO

3

3

 at conditions much more extreme than

 at conditions much more extreme than

those found in Earth’s core–mantle boundary.

those found in Earth’s core–mantle boundary.

These calculations predicted that at pressures of ~11 Mbar,

These calculations predicted that at pressures of ~11 Mbar,

MgSiO

MgSiO

3

3

 dissociates in the following way:

 dissociates in the following way:

MgSiO

MgSiO

3

3

 [CaIrO

 [CaIrO

3

3

 structure] 

 structure] 

→

→

 MgO [CsCl structure]

 MgO [CsCl structure]

                                         + SiO

                                         + SiO

2

2

 [cotunnite structure].

 [cotunnite structure].

7

•

These transformations in the structures of MgO and SiO

These transformations in the structures of MgO and SiO

2

2

allow an important connection to be made 

allow an important connection to be made 

between DFT

between DFT

calculations and experiments

calculations and experiments

 since these transformations

 since these transformations

occur at conditions that can be directly probed in

occur at conditions that can be directly probed in

laboratory experiments.

laboratory experiments.

•

The transition pressures  predicted using DFT and

The transition pressures  predicted using DFT and

observed experimentally are in good agreement, giving a

observed experimentally are in good agreement, giving a

strong indication of the accuracy of these calculations.

strong indication of the accuracy of these calculations.

8

1.2 THE  SCHR

1.2 THE  SCHR

ö

ö

DINGER EQUATION

DINGER EQUATION

•

Since the 

Since the 

mass

mass

 of the nucleus is much 

 of the nucleus is much 

larger

larger

 than the

 than the

mass of the electron, the 

mass of the electron, the 

response

response

 of electrons to

 of electrons to

environmental changes is much 

environmental changes is much 

faster

faster

 than that of

 than that of

nucleus.

nucleus.

•

As a result, we can split our physical question into 

As a result, we can split our physical question into 

two

two

pieces

pieces

. First, we solve, for fixed positions of the atomic

. First, we solve, for fixed positions of the atomic

nuclei, the equations that describe the electron motion.

nuclei, the equations that describe the electron motion.

For a given set of electrons moving in the field of a set of

For a given set of electrons moving in the field of a set of

nuclei, we find the lowest energy configuration, or state,

nuclei, we find the lowest energy configuration, or state,

of the electrons.

of the electrons.

9

If we have M nuclei at positions R1, . . . , RM,

If we have M nuclei at positions R1, . . . , RM,

we can express the ground-state energy, E, as a function of

we can express the ground-state energy, E, as a function of

the positions of these nuclei, E(R1, . . . , RM)

the positions of these nuclei, E(R1, . . . , RM)

we are able to calculate this potential energy surface we can

we are able to calculate this potential energy surface we can

tackle the original problem posed above—

tackle the original problem posed above—

how does the

how does the

energy of the material change as we move its atoms

energy of the material change as we move its atoms

around?

around?

10

11

Here, m is the electron mass. For the Hamiltonian we have

Here, m is the electron mass. For the Hamiltonian we have

chosen, c is the electronic wave function, which is a function

chosen, c is the electronic wave function, which is a function

of each of the spatial coordinates of each of the N electrons,

of each of the spatial coordinates of each of the N electrons,

so c ¼ c(r1, . . . , rN), and E is the groundstate energy of the

so c ¼ c(r1, . . . , rN), and E is the groundstate energy of the

electrons.

electrons.

Schrödinger equation under the interaction system of multiple 

Schrödinger equation under the interaction system of multiple 

electrons and polyatomic nucleus

electrons and polyatomic nucleus

：

：

The quantity of physical interest is really the probability

The quantity of physical interest is really the probability

that a set of N electrons in any order have coordinates r1, . .

that a set of N electrons in any order have coordinates r1, . .

.rN. A closely related quantity is the density of electrons at

.rN. A closely related quantity is the density of electrons at

a particular position in space, n(r). This can be written in

a particular position in space, n(r). This can be written in

terms of the individual electron wave functions as

terms of the individual electron wave functions as

The point of this discussion is that the electron density, 

The point of this discussion is that the electron density, 

n(r)

n(r)

contains a great amount of the information 

contains a great amount of the information 

that is actually

that is actually

physically observable from the full wave function solution

physically observable from the full wave function solution

to the Schrödinger equation.

to the Schrödinger equation.

12

1.3 DENSITY FUNCTIONAL THEORY—FROM WAVE

1.3 DENSITY FUNCTIONAL THEORY—FROM WAVE

FUNCTIONS TO ELECTRON DENSITY

FUNCTIONS TO ELECTRON DENSITY

The entire field of density functional theory rests on 

The entire field of density functional theory rests on 

two

two

fundamental mathematical theorems 

fundamental mathematical theorems 

proved by Kohn and

proved by Kohn and

Hohenberg and the derivation of 

Hohenberg and the derivation of 

a set of equations

a set of equations

 by

 by

Kohn and Sham.

Kohn and Sham.

The first theorem is: 

The first theorem is: 

The ground-state energy from

The ground-state energy from

Schrödinger’s equation is a unique functional of

Schrödinger’s equation is a unique functional of

the electron density.

the electron density.

13

This theorem states that there exists a one-to-one mapping

This theorem states that there exists a one-to-one mapping

between the ground-state wave function and the ground-

between the ground-state wave function and the ground-

state electron density.

state electron density.

we can restate Hohenberg and Kohn’s result by saying

we can restate Hohenberg and Kohn’s result by saying

that the ground-state energy E can be expressed as E[n(r)],

that the ground-state energy E can be expressed as E[n(r)],

where n(r) is the electron density.

where n(r) is the electron density.

Another way to restate Hohenberg and Kohn’s result is

Another way to restate Hohenberg and Kohn’s result is

that the ground-state electron density uniquely determines

that the ground-state electron density uniquely determines

all properties, including the energy and wave function, of

all properties, including the energy and wave function, of

the ground state.

the ground state.

14

Why is this result important?

Why is this result important?

It means that we can think about solving the

It means that we can think about solving the

Schrödinger equation by finding a function of

Schrödinger equation by finding a function of

three spatial variables, the electron density,

three spatial variables, the electron density,

rather than a function of 3N variables, the

rather than a function of 3N variables, the

wave function.

wave function.

15

The second Hohenberg–Kohn theorem defines an

The second Hohenberg–Kohn theorem defines an

important property of the functional: 

important property of the functional: 

The electron

The electron

density that minimizes the energy of the overall

density that minimizes the energy of the overall

functional is the true electron density corresponding

functional is the true electron density corresponding

to the full solution of the Schrödinger equation.

to the full solution of the Schrödinger equation.

16

The energy functional can be written as 

The energy functional can be written as 

：

：

Where we have split the functional into a collection of

Where we have split the functional into a collection of

terms we can write down in a simple analytical form,

terms we can write down in a simple analytical form,

E

E

known

known

[{

[{

Ψ

i

i

}], and everything else, E

}], and everything else, E

XC

XC

. The“known” terms

. The“known” terms

include four contributions:

include four contributions:

17

The terms on the right are, in order, 

The terms on the right are, in order, 

the electron kinetic

the electron kinetic

energies

energies

, 

, 

the Coulomb interactions between the electrons

the Coulomb interactions between the electrons

and the nuclei

and the nuclei

, 

, 

the Coulomb interactions between pairs of

the Coulomb interactions between pairs of

electrons

electrons

, 

, 

and the Coulomb interactions between pairs of

and the Coulomb interactions between pairs of

Nuclei

Nuclei

.

.

The other term in the complete energy functional, 

The other term in the complete energy functional, 

E

E

XC

XC

[{

[{

Ψ

Ψ

i

i

}],

}],

is the exchange–correlation functional

is the exchange–correlation functional

, and it is defined to

, and it is defined to

include all the quantum mechanical effects that are not

include all the quantum mechanical effects that are not

included in the “known” terms.

included in the “known” terms.

18

Let us imagine for now that we can express the as-yet-

Let us imagine for now that we can express the as-yet-

undefined exchange–correlation energy functional in some

undefined exchange–correlation energy functional in some

useful way. 

useful way. 

What is involved in finding minimum energy

What is involved in finding minimum energy

solutions of the total energy functional?

solutions of the total energy functional?

This difficulty was solved by Kohn and Sham, who showed

This difficulty was solved by Kohn and Sham, who showed

that 

that 

the task of finding the right electron density 

the task of finding the right electron density 

can be

can be

expressed in a way that involves solving a set of equations

expressed in a way that involves solving a set of equations

in which each equation 

in which each equation 

only involves a single electron

only involves a single electron

.

.

19

The Kohn–Sham equations have the form

The Kohn–Sham equations have the form

On the left-hand side of the Kohn–Sham equations there

On the left-hand side of the Kohn–Sham equations there

are three potentials, 

are three potentials, 

V

V

, 

, 

V

V

H

H

, and 

, and 

V

V

XC

XC

.

.

20

V 

V 

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21

V

V

H

H

The second is called the Hartree potential and is defined by

The second is called the Hartree potential and is defined by

The Hartree potential includes a socalled self-interaction

The Hartree potential includes a socalled self-interaction

contribution because the electron we are describing in the

contribution because the electron we are describing in the

Kohn–Sham equation is also part of the total electron

Kohn–Sham equation is also part of the total electron

density, so part of V

density, so part of V

H

H

 involves a Coulomb interaction

 involves a Coulomb interaction

between the electron and itself.

between the electron and itself.

22

V

V

XC

XC

can formally be defined as a “functional derivative”

can formally be defined as a “functional derivative”

of the exchange–correlation energy:

of the exchange–correlation energy:

23

If you have a vague sense that there is something

If you have a vague sense that there is something

circular about our discussion of the Kohn–Sham

circular about our discussion of the Kohn–Sham

equations you are exactly right.

equations you are exactly right.

To break this circle, the problem is usually treated in

To break this circle, the problem is usually treated in

an iterative way as outlined in the following

an iterative way as outlined in the following

algorithm:

algorithm:

24

25

4. Compare the calculated electron density, n

4. Compare the calculated electron density, n

KS

KS

(r),

(r),

with the electron density used in solving the Kohn–

with the electron density used in solving the Kohn–

Sham equations, n(r). If the two densities are the

Sham equations, n(r). If the two densities are the

same, then this is the ground-state electron density,

same, then this is the ground-state electron density,

and it can be used to compute the total energy. If

and it can be used to compute the total energy. If

the two densities are different, then the trial

the two densities are different, then the trial

electron density must be updated in some way.

electron density must be updated in some way.

Once this is done, the process begins again from

Once this is done, the process begins again from

step 2.

step 2.

26

1.4 EXCHANGE–CORRELATION

1.4 EXCHANGE–CORRELATION

FUNCTIONAL

FUNCTIONAL

Local Density Approximation , LDA

Local Density Approximation , LDA

we set the exchange–correlation potential at each position

we set the exchange–correlation potential at each position

to be the known exchange–correlation potential from the

to be the known exchange–correlation potential from the

uniform electron gas at the electron density observed at

uniform electron gas at the electron density observed at

that position:

that position:

This approximation uses only the local density to define the

This approximation uses only the local density to define the

approximate exchange–correlation functional.

approximate exchange–correlation functional.

27

The LDA gives us a way to completely define the Kohn–

The LDA gives us a way to completely define the Kohn–

Sham equations, but it is crucial to remember that the

Sham equations, but it is crucial to remember that the

results from these equations 

results from these equations 

do not exactly solve the true

do not exactly solve the true

Schrödinger equation

Schrödinger equation

 because we are not using the true

 because we are not using the true

exchange–correlation functional.

exchange–correlation functional.

28

It should not surprise you that the 

It should not surprise you that the 

LDA is not the only

LDA is not the only

functional

functional

 that has been tried within DFT calculations. The

 that has been tried within DFT calculations. The

development of functionals that more faithfully represent

development of functionals that more faithfully represent

nature remains one of the most important areas of active

nature remains one of the most important areas of active

research in the quantum chemistry community.

research in the quantum chemistry community.

29

Generalized Gradient Approximation , GGA

Generalized Gradient Approximation , GGA

Two of the most widely used functionals in calculations

Two of the most widely used functionals in calculations

involving solids are the Perdew–Wang functional (PW91)

involving solids are the Perdew–Wang functional (PW91)

and the Perdew–Burke–Ernzerhof functional (PBE). Each of

and the Perdew–Burke–Ernzerhof functional (PBE). Each of

these functionals are GGA functionals, and dozens of other

these functionals are GGA functionals, and dozens of other

GGA functionals have been developed and used,

GGA functionals have been developed and used,

particularly for calculations with isolated molecules.

particularly for calculations with isolated molecules.

30

1.5 THE QUANTUM CHEMISTRY

1.5 THE QUANTUM CHEMISTRY

TOURIST

TOURIST

As you read about the approaches aside from DFT that exist

As you read about the approaches aside from DFT that exist

for finding numerical solutions of the Schrödinger

for finding numerical solutions of the Schrödinger

equation, it is likely that you will rapidly encounter a

equation, it is likely that you will rapidly encounter a

bewildering array of acronyms.

bewildering array of acronyms.

This section aims to present an overview of quantum

This section aims to present an overview of quantum

chemical methods on the level of a phrase book or travel

chemical methods on the level of a phrase book or travel

guide.

guide.

31

1.5.1 Localized and Spatially Extended Functions

1.5.1 Localized and Spatially Extended Functions

As an example of a spatially localized function, Fig. 1.1

As an example of a spatially localized function, Fig. 1.1

shows the function

shows the function

32

33

We could include more information in this final function by

We could include more information in this final function by

including more individual functions within its definition.

including more individual functions within its definition.

Also, we could build up functions that describe multiple

Also, we could build up functions that describe multiple

atoms simply by using an appropriate set of localized

atoms simply by using an appropriate set of localized

functions for each individual atom.

functions for each individual atom.

34

What if we are interested in a bulk material such as the

What if we are interested in a bulk material such as the

atoms in solid silicon or the atoms beneath the surface of a

atoms in solid silicon or the atoms beneath the surface of a

metal catalyst?

metal catalyst?

A useful alternative is to use periodic functions to describe

A useful alternative is to use periodic functions to describe

the wave functions or electron densities.

the wave functions or electron densities.

35

36

Figure 1.2 shows a simple example of this idea by plotting:

Figure 1.2 shows a simple example of this idea by plotting:

The resulting function is periodic; that is

The resulting function is periodic; that is

37

1.5.2 Hartree–Fock Method

1.5.2 Hartree–Fock Method

Suppose we would like to approximate the wave function of

Suppose we would like to approximate the wave function of

N electrons. Let us assume for the moment that the

N electrons. Let us assume for the moment that the

electrons have no effect on each other. If this is true, the

electrons have no effect on each other. If this is true, the

Hamiltonian for the electrons may be written as

Hamiltonian for the electrons may be written as

：

：

where h

where h

i

i

 describes the kinetic and potential energy of

 describes the kinetic and potential energy of

electron i.

electron i.

38

If we write down the Schrödinger equation for just one

If we write down the Schrödinger equation for just one

electron based on this Hamiltonian, the solutions would

electron based on this Hamiltonian, the solutions would

satisfy :

satisfy :

The eigenfunctions defined by this equation are called 

The eigenfunctions defined by this equation are called 

spin

spin

orbitals

orbitals

.

.

39

When the total Hamiltonian is simply a sum of one-electron

When the total Hamiltonian is simply a sum of one-electron

operators, h

operators, h

i

i

, it follows that the eigenfunctions of H are

, it follows that the eigenfunctions of H are

products of the one-electron spin orbitals:

products of the one-electron spin orbitals:

The energy of this wave function is the sum of the spin

The energy of this wave function is the sum of the spin

orbital energies, E = E

orbital energies, E = E

j1

j1

+…+E

+…+E

jN

jN

 .

 .

40

Electrons are fermions, the wave function must change sign

Electrons are fermions, the wave function must change sign

if two electrons change places with each other. Exchanging

if two electrons change places with each other. Exchanging

two electrons does not change the sign of the Hartree

two electrons does not change the sign of the Hartree

product, which is a serious drawback.

product, which is a serious drawback.

We can obtain a better approximation to the wave function

We can obtain a better approximation to the wave function

by using 

by using 

a Slater determinant

a Slater determinant

.

.

41

42

In a Hartree–Fock (HF) calculation, we fix the positions of

In a Hartree–Fock (HF) calculation, we fix the positions of

the atomic nuclei and aim to determine the wave function

the atomic nuclei and aim to determine the wave function

of N-interacting electrons. The Schrödinger equation for

of N-interacting electrons. The Schrödinger equation for

each electron is written as

each electron is written as

：

：

43

How to express the solution of the above single

How to express the solution of the above single

electron equation? How do you combine these

electron equation? How do you combine these

solutions into N electron wave functions?

solutions into N electron wave functions?

44

To do this, we define a finite set of functions that can be

To do this, we define a finite set of functions that can be

added together to approximate the exact spin orbitals. If

added together to approximate the exact spin orbitals. If

our finite set of functions is written as 

our finite set of functions is written as 

Φ

Φ

1

1

(x), 

(x), 

Φ

Φ

2

2

(x),…,

(x),…,

Φ

Φ

K

K

(x),

(x),

then we can approximate the spin orbitals as:

then we can approximate the spin orbitals as:

45

When using this expression, we only need to find the

When using this expression, we only need to find the

expansion coefficients, a

expansion coefficients, a

j,i

j,i

, for i 1, . . . , K and j 1, . . . , N to

, for i 1, . . . , K and j 1, . . . , N to

fully define all the spin orbitals that are used in the HF

fully define all the spin orbitals that are used in the HF

method. The set of functions 

method. The set of functions 

Φ

Φ

1

1

(x), 

(x), 

Φ

Φ

2

2

(x), . . . , 

(x), . . . , 

Φ

Φ

K

K

(x) is

(x) is

called the 

called the 

basis set 

basis set 

for the calculation.

for the calculation.

46

Like solving the Kohn-Sham equation, an HF calculation is

Like solving the Kohn-Sham equation, an HF calculation is

an iterative procedure that can be outlined as follows:

an iterative procedure that can be outlined as follows:

1.

Make an initial estimate of the spin orbitals

Make an initial estimate of the spin orbitals

by specifying the expansion coefficients, a

by specifying the expansion coefficients, a

j,i.

j,i.

2. From the current estimate of the spin orbitals, define the

2. From the current estimate of the spin orbitals, define the

electron density, n(r`):

electron density, n(r`):

3. Using the electron density from step 2, solve the single-

3. Using the electron density from step 2, solve the single-

electron equations for the spin orbitals.

electron equations for the spin orbitals.

47

4. If the spin orbitals found in step 3 are consistent with

4. If the spin orbitals found in step 3 are consistent with

orbitals used in step 2, then these are the solutions to the

orbitals used in step 2, then these are the solutions to the

HF problem we set out to calculate. If not, then a new

HF problem we set out to calculate. If not, then a new

estimate for the spin orbitals must be made and we then

estimate for the spin orbitals must be made and we then

return to step 2.

return to step 2.

48

1

1

.

.

5

5

.

.

4

4

B

B

e

e

y

y

o

o

n

n

d

d

H

H

a

a

r

r

t

t

r

r

e

e

e

e

–

–

F

F

o

o

c

c

k

k

How do more advanced quantum chemical approaches

How do more advanced quantum chemical approaches

improve on the HF method?

improve on the HF method?

The approaches vary, but the common goal is to

The approaches vary, but the common goal is to

include a description of electron correlation.

include a description of electron correlation.

Methods based on a single reference determinant

Methods based on a single reference determinant

are formally known as “post–Hartree–Fock”

are formally known as “post–Hartree–Fock”

methods.

methods.

49

The classification of wave-function-based methods has two

The classification of wave-function-based methods has two

distinct components: 

distinct components: 

the level of theory 

the level of theory 

and 

and 

the basis set

the basis set

.

.

To illustrate the role of the level of theory and the basis set,

To illustrate the role of the level of theory and the basis set,

we will look at two properties of a molecule of CH

we will look at two properties of a molecule of CH

4

4

, the C–H

, the C–H

bond length and the ionization energy. Experimentally, the

bond length and the ionization energy. Experimentally, the

C–H bond length is 1.094A˚ and the ionization energy for

C–H bond length is 1.094A˚ and the ionization energy for

methane is 12.61 eV.

methane is 12.61 eV.

50

51

52

The classification of wave-function-based methods has two

The classification of wave-function-based methods has two

distinct components: 

distinct components: 

the level of theory 

the level of theory 

and 

and 

the basis set

the basis set

.

.

To illustrate the role of the level of theory and the basis set,

To illustrate the role of the level of theory and the basis set,

we will look at two properties of a molecule of CH

we will look at two properties of a molecule of CH

4

4

, the C–H

, the C–H

bond length and the ionization energy. Experimentally, the

bond length and the ionization energy. Experimentally, the

C–H bond length is 1.094A˚ and the ionization energy for

C–H bond length is 1.094A˚ and the ionization energy for

methane is 12.61 eV.

methane is 12.61 eV.

53

1.7 WHAT CAN DFT NOT DO?

1.7 WHAT CAN DFT NOT DO?

There are some important situations for which DFT cannot

There are some important situations for which DFT cannot

be expected to be physically accurate.

be expected to be physically accurate.

①

The first situation where DFT calculations have limited

The first situation where DFT calculations have limited

accuracy is in the calculation of electronic excited states.

accuracy is in the calculation of electronic excited states.

②

The second case is that the bandgap of the

The second case is that the bandgap of the

semiconductor and insulator materials obtained in the

semiconductor and insulator materials obtained in the

DFT calculation is underestimated.

DFT calculation is underestimated.

54

③

The weak attraction of van der Waals (vdW)

The weak attraction of van der Waals (vdW)

between atoms and molecules leads to

between atoms and molecules leads to

inaccuracies in calculation results.

inaccuracies in calculation results.

④

The atomic magnitude in the calculation cannot

The atomic magnitude in the calculation cannot

match the actual atomic magnitude.

match the actual atomic magnitude.

55