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  • Numerical Methods for Kohn-Sham Density Functional Theory

    Jianfeng Lu (้ฒๅ‰‘้”‹)

    Duke University jianfeng@math.duke.edu

    January 2019, Banff Workshop on โ€œOptimal Transport Methods in Density Functional Theoryโ€

  • Reference Lin Lin, L., and Lexing Ying, Numerical Methods for Kohn-Sham Density Functional Theory, Acta Numerica, 2019 (under review)

  • Electronic structure theory

    We consider the electronic structure, which is given by the many-body electronic Schrรถdinger equation

    (โˆ’ 1 2

    ๐‘

    โˆ‘ ๐‘–=1

    ฮ”๐‘ฅ๐‘– + ๐‘

    โˆ‘ ๐‘–=1

    ๐‘‰ext(๐‘ฅ๐‘–) + โˆ‘ 1โ‰ค๐‘–

  • Electronic structure theory

    We consider the electronic structure, which is given by the many-body electronic Schrรถdinger equation

    (โˆ’ 1 2

    ๐‘

    โˆ‘ ๐‘–=1

    ฮ”๐‘ฅ๐‘– + ๐‘

    โˆ‘ ๐‘–=1

    ๐‘‰ext(๐‘ฅ๐‘–) + โˆ‘ 1โ‰ค๐‘–

  • Solving electronic Schrรถdinger equations: โ€ข Tight-binding approximations (LCAO) โ€ข Density functional theory

    Orbital-free DFT Kohn-Sham DFT

    โ€ข Wavefunction methods Hartree-Fock Mรธller-Plesset perturbation theory Configuration interaction Coupled cluster Multi-configuration self-consistent field Neural network ansatz

    โ€ข GW approximation; Bethe-Salpeter equation โ€ข Quantum Monte Carlo (VMC, DMC, etc.) โ€ข Density matrix renormalization group (DMRG) / tensor networks โ€ข ...

  • Solving electronic Schrรถdinger equations: โ€ข Tight-binding approximations (LCAO) โ€ข Density functional theory

    Orbital-free DFT Kohn-Sham DFT

    10 of 18 most cited papers in physics [Perdew 2010] More than 50, 000 citations on Google scholar

    Figure: DFT papers count [Burke 2012].

  • Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: View energy as a functional of the one-body electron density ๐œŒ โˆถ โ„3 โ†’ โ„โ‰ฅ0:

    ๐œŒ(๐‘ฅ) = ๐‘ โˆซ|ฮจ| 2(๐‘ฅ, ๐‘ฅ2, โ‹ฏ , ๐‘ฅ๐‘ ) d๐‘ฅ2 โ‹ฏ d๐‘ฅ๐‘ .

    Levy-Lieb variational principle [Levy 1979, Lieb 1983]:

    ๐ธ0 = infฮจ โŸจฮจ, ๐ปฮจโŸฉ = inf๐œŒ infฮจ,ฮจโ†ฆ๐œŒโŸจฮจ, ๐ปฮจโŸฉ = inf๐œŒ ๐ธ(๐œŒ).

    The energy functional has the form

    ๐ธ(๐œŒ) = ๐‘‡๐‘ (๐œŒ) + โˆซ ๐œŒ๐‘‰ext + 1 2 โˆฌ

    ๐œŒ(๐‘ฅ)๐œŒ(๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| + ๐ธxc(๐œŒ)

    ๐‘‡๐‘ (๐œŒ): Kinetic energy of non-interacting electrons; ๐ธxc(๐œŒ): Exchange-correlation energy, which encodes the many-body interaction between electrons (chemistry).

  • Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: View energy as a functional of the one-body electron density ๐œŒ โˆถ โ„3 โ†’ โ„โ‰ฅ0:

    ๐œŒ(๐‘ฅ) = ๐‘ โˆซ|ฮจ| 2(๐‘ฅ, ๐‘ฅ2, โ‹ฏ , ๐‘ฅ๐‘ ) d๐‘ฅ2 โ‹ฏ d๐‘ฅ๐‘ .

    Levy-Lieb variational principle [Levy 1979, Lieb 1983]:

    ๐ธ0 = infฮจ โŸจฮจ, ๐ปฮจโŸฉ = inf๐œŒ infฮจ,ฮจโ†ฆ๐œŒโŸจฮจ, ๐ปฮจโŸฉ

    = inf ๐œŒ

    ๐ธ(๐œŒ).

    The energy functional has the form

    ๐ธ(๐œŒ) = ๐‘‡๐‘ (๐œŒ) + โˆซ ๐œŒ๐‘‰ext + 1 2 โˆฌ

    ๐œŒ(๐‘ฅ)๐œŒ(๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| + ๐ธxc(๐œŒ)

    ๐‘‡๐‘ (๐œŒ): Kinetic energy of non-interacting electrons; ๐ธxc(๐œŒ): Exchange-correlation energy, which encodes the many-body interaction between electrons (chemistry).

  • Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: View energy as a functional of the one-body electron density ๐œŒ โˆถ โ„3 โ†’ โ„โ‰ฅ0:

    ๐œŒ(๐‘ฅ) = ๐‘ โˆซ|ฮจ| 2(๐‘ฅ, ๐‘ฅ2, โ‹ฏ , ๐‘ฅ๐‘ ) d๐‘ฅ2 โ‹ฏ d๐‘ฅ๐‘ .

    Levy-Lieb variational principle [Levy 1979, Lieb 1983]:

    ๐ธ0 = infฮจ โŸจฮจ, ๐ปฮจโŸฉ = inf๐œŒ infฮจ,ฮจโ†ฆ๐œŒโŸจฮจ, ๐ปฮจโŸฉ = inf๐œŒ ๐ธ(๐œŒ).

    The energy functional has the form

    ๐ธ(๐œŒ) = ๐‘‡๐‘ (๐œŒ) + โˆซ ๐œŒ๐‘‰ext + 1 2 โˆฌ

    ๐œŒ(๐‘ฅ)๐œŒ(๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| + ๐ธxc(๐œŒ)

    ๐‘‡๐‘ (๐œŒ): Kinetic energy of non-interacting electrons; ๐ธxc(๐œŒ): Exchange-correlation energy, which encodes the many-body interaction between electrons (chemistry).

  • Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: View energy as a functional of the one-body electron density ๐œŒ โˆถ โ„3 โ†’ โ„โ‰ฅ0:

    ๐œŒ(๐‘ฅ) = ๐‘ โˆซ|ฮจ| 2(๐‘ฅ, ๐‘ฅ2, โ‹ฏ , ๐‘ฅ๐‘ ) d๐‘ฅ2 โ‹ฏ d๐‘ฅ๐‘ .

    Levy-Lieb variational principle [Levy 1979, Lieb 1983]:

    ๐ธ0 = infฮจ โŸจฮจ, ๐ปฮจโŸฉ = inf๐œŒ infฮจ,ฮจโ†ฆ๐œŒโŸจฮจ, ๐ปฮจโŸฉ = inf๐œŒ ๐ธ(๐œŒ).

    The energy functional has the form

    ๐ธ(๐œŒ) = ๐‘‡๐‘ (๐œŒ) + โˆซ ๐œŒ๐‘‰ext + 1 2 โˆฌ

    ๐œŒ(๐‘ฅ)๐œŒ(๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| + ๐ธxc(๐œŒ)

    ๐‘‡๐‘ (๐œŒ): Kinetic energy of non-interacting electrons; ๐ธxc(๐œŒ): Exchange-correlation energy, which encodes the many-body interaction between electrons (chemistry).

  • Kohn-Sham density functional theory Kohn-Sham density functional theory introduces one-particle orbitals to better approximate the kinetic and exchange-correlation energies.

    It is today the most widely used electronic structure theory, which achieves the best compromise between accuracy and cost.

    The energy functional is minimized for ๐‘ orbitals {๐œ“๐‘–} โŠ‚ ๐ป1(โ„3).

    ๐ธKS({๐œ“๐‘–}) = ๐‘

    โˆ‘ ๐‘–=1

    1 2 โˆซ|โˆ‡๐œ“๐‘–|

    2 + โˆซ ๐œŒ๐‘‰ext + 1 2 โˆฌ

    ๐œŒ(๐‘ฅ)๐œŒ(๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| + ๐ธxc(๐œŒ)

    where (we consider โ€œspin-lessโ€ electrons through the talk)

    ๐œŒ(๐‘ฅ) = ๐‘

    โˆ‘ ๐‘–=1

    |๐œ“๐‘–|2(๐‘ฅ).

    Note that ๐ธKS can be still viewed as a functional of ๐œŒ, implicitly.

  • Kohn-Sham density functional theory

    ๐ธKS({๐œ“๐‘–}) = ๐‘

    โˆ‘ ๐‘–=1

    1 2 โˆซ|โˆ‡๐œ“๐‘–|

    2 + โˆซ ๐œŒ๐‘‰ext + 1 2 โˆฌ

    ๐œŒ(๐‘ฅ)๐œŒ(๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| + ๐ธxc

    The exchange-correlation functionals counts for the corrections from the many-body interactions between electrons:

    โ€ข Semi-local exchange-correlation functionals Local density approximation: ๐ธxc = โˆซ ๐‘’xc(๐œŒ(๐‘ฅ)) Generalized gradient approximation:

    ๐ธxc = โˆซ ๐‘’xc(๐œŒ(๐‘ฅ), 1 2 |โˆ‡โˆš๐œŒ(๐‘ฅ)|

    2)

    โ€ข Nonlocal exchange-correlation functionals

  • Kohn-Sham density functional theory

    ๐ธKS({๐œ“๐‘–}) = ๐‘

    โˆ‘ ๐‘–=1

    1 2 โˆซ|โˆ‡๐œ“๐‘–|

    2 + โˆซ ๐œŒ๐‘‰ext + 1 2 โˆฌ

    ๐œŒ(๐‘ฅ)๐œŒ(๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| + ๐ธxc

    โ€ข Nonlocal exchange-correlation functionals e.g., exact exchange + RPA correlation, ๐ธxc = ๐ธ๐‘ฅ + ๐ธ๐‘:

    ๐ธ๐‘ฅ = โˆ’ 1 2

    ๐‘

    โˆ‘ ๐‘–,๐‘—=1 โˆฌ

    ๐œ“โˆ—๐‘– (๐‘ฅ)๐œ“๐‘—(๐‘ฅ)๐œ“๐‘–(๐‘ฆ)๐œ“โˆ—๐‘— (๐‘ฆ) |๐‘ฅ โˆ’ ๐‘ฆ| d๐‘ฅ d๐‘ฆ;

    ๐ธ๐‘ = 1

    2๐œ‹ โˆซ โˆž

    0 tr[ln(1 โˆ’ ๐œ’0(๐‘–๐œ”)๐œˆ) + ๐œ’0(๐‘–๐œ”)๐œˆ] d๐œ”,

    with ๐œˆ the Coulomb operator and ๐œ’0 Kohn-Sham polarizability operator:

    ๐œ’0(๐‘ฅ, ๐‘ฆ, ๐‘–๐œ”) = 2 occ

    โˆ‘ ๐‘—

    unocc

    โˆ‘ ๐‘˜

    ๐œ“โˆ—๐‘— (๐‘ฅ)๐œ“๐‘˜(๐‘ฅ)๐œ“โˆ—๐‘˜ (๐‘ฆ)๐œ“๐‘—(๐‘ฆ) ๐œ€๐‘— โˆ’ ๐œ€๐‘˜ โˆ’ ๐‘–๐œ”

    .

  • Kohn-Sham density functional theory has a similar structure as a mean field type theory (at least for semilocal xc): The electrons interact through an effective potential. The electron density is given by an effective Hamiltonian:

    ๐ปeff(๐œŒ)๐œ“๐‘– = ๐œ€๐‘–๐œ“๐‘–, ๐œŒ(๐‘ฅ) = occ

    โˆ‘ ๐‘–

    |๐œ“๐‘–|2(๐‘ฅ);

    where the first ๐‘ orbitals are occupied (aufbau principle). The effective Hamiltonian captures the interactions of electrons:

    ๐ปeff(๐œŒ) = โˆ’ 1 2ฮ” + ๐‘‰eff(๐œŒ);

    ๐‘‰eff(๐œŒ) = ๐‘‰๐‘(๐œŒ) + ๐‘‰xc(๐œŒ).

    Note that this is a nonlinear eigenvalue problem. We can view it as a fixed-point equation for the density ๐œŒ:

    ๐œŒ = ๐นKS(๐œŒ).

  • Kohn-Sham map Kohn-Sham fixed-point equation

    ๐œŒ = ๐นKS(๐œŒ),

    where ๐นKS is known as the Kohn-Sham map, defined through the eigenvalue problem associated with ๐ปeff(๐œŒ). Given an effective Hamiltonian ๐ปeff, we ask for its low-lying eigenspace: the range of the spectral projection

    ๐‘ƒ = ๐œ’(โˆ’โˆž,๐œ–๐น ](๐ปeff),

    thus ๐œŒ is given as the diagonal of the kernel of the operator ๐‘ƒ .

    It is often more advantageous to represent ๐‘ƒ in terms of Greenโ€™s functions (๐œ† โˆ’ ๐ปeff)โˆ’1, so to turn the eigenvalue problem into solving equations. Let ๐’ž be a contour around the occupied spectrum, we have

    ๐‘ƒ = ๐œ’(โˆ’โˆž,๐œ–๐น ](๐ปeff) = 1

    2๐œ‹๐šค โˆฎ๐’ž (๐œ† โˆ’ ๐ปeff)โˆ’1 d๐œ†.

  • Kohn-Sham map Kohn-Sham fixed-point equation

    ๐œŒ = ๐นKS(๐œŒ),

    where ๐นKS is known as the Kohn-Sham map, defined through the eigenvalue problem associated with ๐ปeff(๐œŒ). Given an effective Hamiltonian ๐ปeff, we ask for its low-lying eigenspace: the range of the spectral projection

    ๐‘ƒ = ๐œ’(โˆ’โˆž,๐œ–๐น ](๐ปeff),

    thus ๐œŒ is given as the diagonal of the kernel of the operator ๐‘ƒ . It is often more advantageous to represent ๐‘ƒ in terms of Greenโ€™s functions (๐œ† โˆ’ ๐ปeff)โˆ’1, so to turn the eigenvalue problem into solving equations. Let ๐’ž be a contour around the occupied spectrum, we have

    ๐‘ƒ = ๐œ’(โˆ’โˆž,๐œ–๐น ](๐ปeff) = 1

    2๐œ‹๐šค โˆฎ๐’ž (๐œ† โˆ’ ๐ปeff)โˆ’1 d๐œ†.

  • Numerical methods

    Kohn-Sham fixed-point equation

    ๐œŒ = ๐นKS(๐œŒ).

    Usually solved numerically based on self-consistent field (SCF) iteration (alternative methods, such as di

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