Kohn-Sham idea

Density functional theory

exact mapping between non-interacting and interacting electronic systems

real � V e↵[⇢(r)]�! non-int.

Hnon-int. =

NX

i

(�1

2r2 + V

e↵)

(�1

2r2 + V e↵)�i(r) = ✏i�i(r)

effective potential exists, so that densities are the same (ground-state only!!)

one-electron wavefunctions: Kohn-Sham orbitals

density of both systems

with exact effective potential

⇢(r) = ⇢real(r)

⇢(r) =nX

i

|�i(r)|2<latexit sha1_base64="J05VE8Rh6BiHXsHOG5kk3E+hzxQ=">AAACFHicdVDLSgMxFM3UV62vqks3wSIowpBpO9guBNGNSwWrQqcOmTTThmYyQ5IRyrQf4cZfceNCEbcu3Pk3prXiAz1w4XDOvdx7T5BwpjRCb1Zuanpmdi4/X1hYXFpeKa6unas4lYQ2SMxjeRlgRTkTtKGZ5vQykRRHAacXQe9o5F9cU6lYLM50P6GtCHcECxnB2kh+cdeT3Xg784Iwk8Phzr6n0shnV2LgJV3msy9ncFX2iyVkI7fuOggi20VOvTIi9Xqt6rrQsdEYJTDBiV989doxSSMqNOFYqaaDEt3KsNSMcDoseKmiCSY93KFNQwWOqGpl46eGcMsobRjG0pTQcKx+n8hwpFQ/CkxnhHVX/fZG4l9eM9VhrZUxkaSaCvKxKEw51DEcJQTbTFKied8QTCQzt0LSxRITbXIsmBA+P4X/k/Oy7VTs8mm1dHA4iSMPNsAm2AYO2AMH4BicgAYg4AbcgQfwaN1a99aT9fzRmrMmM+vgB6yXd/tHn20=</latexit>

ground-state only

E[⇢(r)] = min !⇢

h |H| i � E0

variational principle

energy functional

Kohn-Sham idea

Density functional theory

exact mapping between non-interacting and interacting electronic systems

real � V e↵[⇢(r)]�! non-int.

E[⇢(r)] = T [⇢(r)] + Eee[⇢(r)] + Ene[⇢(r)]<latexit sha1_base64="qQ+EsmawZHFhtwvJ6M1WJHnyrkk=">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</latexit>

E[⇢(r)] = T [⇢(r)] + Eee[⇢(r)] +

Z⇢[(r)]

X

A

ZA

|r�RA|dr

<latexit sha1_base64="X6AUcOsYOHD68mZ5uJ9yYcLZgbo=">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</latexit>

E[⇢(r)] = T [⇢(r)] + Eee[⇢(r)] +

Z⇢[(r)]Vne(r)dr

<latexit sha1_base64="Du+3aF3VkyscY8q2+kUgB7S80iA=">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</latexit>

ground-state only

E[⇢(r)] = min !⇢

h |H| i � E0

variational principle

energy functional for interacting system

Kohn-Sham idea

Density functional theory

exact mapping between non-interacting and interacting electronic systems

real � V e↵[⇢(r)]�! non-int.

E[⇢(r)] = T [⇢(r)] + Eee[⇢(r)] +

Z⇢[(r)]Vne(r)dr

<latexit sha1_base64="Du+3aF3VkyscY8q2+kUgB7S80iA=">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</latexit>

E[⇢(r)] = TS [⇢(r)] + EH

ee[⇢(r)] +

Z⇢[(r)]Vne(r)dr+ EXC [⇢(r)]

<latexit sha1_base64="06UpnZU11hqmO7ksTiXzdSiyS2Q=">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</latexit>

EXC [⇢(r)] = (T [⇢(r)]� TS [⇢(r)]) + (Eee[⇢(r)]� EH

ee[⇢(r)])

<latexit sha1_base64="r/zhriDKzZQplzSIDJ1zsjUtFvo=">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</latexit>

add and subtract energy functionals for a non interacting systems

dump all that we don’t want to compute in the exchange-correlation functional

Density functional theory

electron-electron repulsion: Hartree potential

electron-nuclei attraction: external potential

exchange-correlation energy (e.g. uniform electron gas)

Een[⇢] =

Z NX

A

ZA

|RA � r|⇢(r)dr =

ZVen(r)⇢(r)dr

kinetic energy of non-interacting systems

energy functional

ELDAXC [⇢] =

Z✏XC(⇢(r))⇢(r)dr

TS [⇢(r)] = min !⇢

h |TS | i<latexit sha1_base64="YpXVFXO7+mGDWCFNOL/f8RyegC4=">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</latexit>

EH

ee[⇢(r)] =

1

2

Z Z⇢(r)⇢(r0)

|r� r0| drdr0

<latexit sha1_base64="DWjJ037kdW84ZSx11xL94NMLbdU=">AAACZnicdVFda9swFJXdfbTZ1qUtYw97EQuj7cOMnNa0eRiUjkEfO1jaQuwFWbluRGXZSHIhqPqTe9vzXvYzqrhJWcZ24cK555yLpKO8FlwbQn4G4dqTp8+er290Xrx8tfm6u7V9oatGMRiySlTqKqcaBJcwNNwIuKoV0DIXcJnffJ7rl7egNK/kNzOrISvpteQFZ9R4atx1X8YWwH23Z26Uqmm1Z9O8sMq5/Qx/wmmhKLOxs32XcmnabqlV68q0u+/s3XL4+MjeuckSP4LdcbdHIpIMkphgEiUkHhzMwWBwfJgkOI5IWz20qPNx90c6qVhTgjRMUK1HMalNZqkynAlwnbTRUFN2Q69h5KGkJejMtjE5/MEzE1xUyrc0uGX/3LC01HpW5t5ZUjPVf2tz8l/aqDHFcWa5rBsDkj0cVDQCmwrPM8cTroAZMfOAMsX9XTGbUp+j8T/T8SEsX4r/Dy76UXwQ9b8e9k5OF3Gso3foPdpDMTpCJ+gMnaMhYuhXsBFsBzvB73AzfBO+fbCGwWJnB61UiO8BZbm8jg==</latexit>

E[⇢(r)] = TS [⇢(r)] + EH

ee[⇢(r)] + Ene[⇢(r)] + EXC [⇢(r)]

<latexit sha1_base64="PvDQh8sCHHqh5CrAeHuWMIRX4O8=">AAACWXicdZFRSyMxFIUz4+p266rVPu5L2CK4LAwZdVj7IMgWwUeXtVpoZ4dMeqcNZjJDkhFKmD/pgyD7V3wwrRVcdS8EDt+5l5ucpKXg2hBy7/krH1bXPjY+Ndc/b2xutbZ3LnVRKQZ9VohCDVKqQXAJfcONgEGpgOapgKv0ujf3r25AaV7ICzMrIc7pRPKMM2ocSlrl6XCkpsWeHaWZVXX9LcbH+CL5/Zp+P/1jz+rEAtRvrcTK9/Gg9wYnrQ4JSNSNQoJJEJGwezAX3e7RYRThMCCL6qBlnSet29G4YFUO0jBBtR6GpDSxpcpwJqBujioNJWXXdAJDJyXNQcd2kUyNdx0Z46xQ7kiDF/TlhKW51rM8dZ05NVP92pvD97xhZbKj2HJZVgYke1qUVQKbAs9jxmOugBkxc4Iyxd1dMZtSRZlxn9F0ITy/FP9fXO4H4UGw/+uwc/JzGUcDfUFf0R4K0Q90gs7QOeojhu7Qg7fqrXl/fc9v+M2nVt9bzrTRP+W3HwEyVbVa</latexit>

Density functional theory

electron-electron repulsion (Hartree potential)

electron-nuclei attraction: external potential

exchange-correlation energy (e.g., uniform electron gas)

minimize functional with respect to density

Vee([⇢], r) =1

2

Z⇢(r0)

|r� r|dr0

Ven([⇢], r) =X

A

ZA

|RA � r|

V LDAXC ([⇢], r) =

@

@⇢ELDA

XC = ⇢(r)@✏XC(⇢(r))

@⇢

��⇢=⇢(r) + ✏XC(⇢(r))

@E =

Z ✓@

@⇢T + Vee + Ene + VXC

◆@⇢dr = 0V H

ee<latexit sha1_base64="UcJ5X61usy1hgzLbWDfec1KacRM=">AAAB73icdVDLTgJBEJzFF+IL9ehlIjHxtJkFNsKN6IUjJvJIAMns0MCE2YczsyZkw0948aAxXv0db/6Ns4CJGq2kk0pVd7q7vEhwpQn5sDJr6xubW9nt3M7u3v5B/vCopcJYMmiyUISy41EFggfQ1FwL6EQSqO8JaHvTq9Rv34NUPAxu9CyCvk/HAR9xRrWROq1BAjC/rQ/yBWITt+o6BBPbJU61lJJqtVJ2XezYZIECWqExyL/3hiGLfQg0E1SprkMi3U+o1JwJmOd6sYKIsikdQ9fQgPqg+sni3jk+M8oQj0JpKtB4oX6fSKiv1Mz3TKdP9UT99lLxL68b61Gln/AgijUEbLloFAusQ5w+j4dcAtNiZghlkptbMZtQSZk2EeVMCF+f4v9Jq2g7Jbt4XS7ULldxZNEJOkXnyEEXqIbqqIGaiCGBHtATerburEfrxXpdtmas1cwx+gHr7RNnvpA5</latexit>

V H

ee<latexit sha1_base64="UcJ5X61usy1hgzLbWDfec1KacRM=">AAAB73icdVDLTgJBEJzFF+IL9ehlIjHxtJkFNsKN6IUjJvJIAMns0MCE2YczsyZkw0948aAxXv0db/6Ns4CJGq2kk0pVd7q7vEhwpQn5sDJr6xubW9nt3M7u3v5B/vCopcJYMmiyUISy41EFggfQ1FwL6EQSqO8JaHvTq9Rv34NUPAxu9CyCvk/HAR9xRrWROq1BAjC/rQ/yBWITt+o6BBPbJU61lJJqtVJ2XezYZIECWqExyL/3hiGLfQg0E1SprkMi3U+o1JwJmOd6sYKIsikdQ9fQgPqg+sni3jk+M8oQj0JpKtB4oX6fSKiv1Mz3TKdP9UT99lLxL68b61Gln/AgijUEbLloFAusQ5w+j4dcAtNiZghlkptbMZtQSZk2EeVMCF+f4v9Jq2g7Jbt4XS7ULldxZNEJOkXnyEEXqIbqqIGaiCGBHtATerburEfrxXpdtmas1cwx+gHr7RNnvpA5</latexit>

TS<latexit sha1_base64="ay0GGC6x3zKUm4X0XvUIc4GUm5U=">AAAB6nicdVDLSgMxFM3UV62vqks3wSK4GjJtBzu7ohuXlT6hHUomTdvQTGZIMkIZ+gluXCji1i9y59+YaSuo6IELh3Pu5d57gpgzpRH6sHIbm1vbO/ndwt7+weFR8fiko6JEEtomEY9kL8CKciZoWzPNaS+WFIcBp91gdpP53XsqFYtES89j6od4ItiYEayN1GwNm8NiCdnI9VwHQWS7yPEqGfG8WtV1oWOjJUpgjcaw+D4YRSQJqdCEY6X6Doq1n2KpGeF0URgkisaYzPCE9g0VOKTKT5enLuCFUUZwHElTQsOl+n0ixaFS8zAwnSHWU/Xby8S/vH6ixzU/ZSJONBVktWiccKgjmP0NR0xSovncEEwkM7dCMsUSE23SKZgQvj6F/5NO2XYqdvmuWqpfr+PIgzNwDi6BA65AHdyCBmgDAibgATyBZ4tbj9aL9bpqzVnrmVPwA9bbJ3n7jfA=</latexit>

Density functional theoryenergy functional of non-interacting system

minimize functional of non-interacting system

kinetic energy functional

effective potential function of true potentials

Enon-int.[⇢] = TS [⇢] +

ZV e↵.(r)⇢(r)dr

<latexit sha1_base64="k2oVDSqFngRE2ww3CVPlyuov+0U=">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</latexit>

�Enon-int.[⇢] =

Z ✓@

@⇢TS [⇢] + V e↵.(r)

◆�⇢(r)dr = 0

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@

@⇢TS [⇢] = �V e↵.(r)

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V e↵.(r) = Ven(r) + V H

ee([⇢], r) + VXC([⇢], r)

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Density functional theoryPractical Kohn-Sham DFT scheme

Step 2: construct effective potential with chosen exchange-correlation potential

Step 3a: solve the Schrödinger equation for non-interacting electrons

(�1

2r2 + V e↵)�i(r) = ✏i�i(r)

Step 3b: compute new density & kinetic energy

Step 4: goto step 2, until convergence is reached

Step 1: guess density ⇢(r)

final energy:

V e↵.(r) = Ven(r) + V H

ee([⇢], r) + VXC([⇢], r)

<latexit sha1_base64="Lbhhw6GGh9Pu490JAggo3w//Nd0=">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</latexit>

⇢(r) =nX

i

|�i(r)|2<latexit sha1_base64="J05VE8Rh6BiHXsHOG5kk3E+hzxQ=">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</latexit>

TS [⇢] =nX

i

h�i|�1

2r2|�ii =

nX

i

✏i �Z

V e↵.(r)⇢(r)dr<latexit sha1_base64="A/POdM+ZTwEnHyAxmHZla/C4bdI=">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</latexit>

E[⇢0] =nX

i

✏i �Z

V e↵.(r)⇢0(r)dr+ Een[⇢0] + EH

ee[⇢0] + EXC [⇢0]

<latexit sha1_base64="SWok/pKkz1VP1xV9Uqf3py1Ybd8=">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</latexit>

Density functional theoryExchange-correlation functionals (“Jacob’s ladder”)

Local Density Approximation

ELDAXC [⇢(r)] =

Zf [⇢(r)]dr

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Generalized Gradient Approximation (GGA)

Meta-GGA

Hybrid (with fraction of Hartree Fock exchange)

Range-separated (fraction of HF exchange depends on distance)

EGGAXC [⇢(r)] =

Zf [⇢(r),r⇢(r)]dr

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EMeta-GGAXC [⇢(r)] =

Zf [⇢(r),r⇢(r),r2⇢(r)]dr

<latexit sha1_base64="j2yMAnIZST+g/VKWTR/OEXFR7rQ=">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</latexit>

VWN

PW91, PBE, BLYP

M06-L

PBE0, B3LYP

EHybrid

XC [⇢(r)] =

Zf [⇢(r),r⇢(r)]dr+ ↵EHF

X<latexit sha1_base64="yCD8rRrWzqdJfIs/EurWLjjT904=">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</latexit>

CAM-B3LYP, LC-ωPBE