Exchange-correlation functionals

Functionals are Equinox modules selected by instance (PBE()) or by name (functional("wb97m-v"), case- and punctuation-insensitive), spanning LDA, GGA, meta-GGA, global-hybrid, and range-separated-hybrid rungs. The Fock matrix is autodiffed from the energy, so adding a functional means writing ε_xc only; range-separated hybrids declare (hf_coeff, hf_coeff_lr, omega) and the engine builds the attenuated K_lr integrals, and a functional declaring nlc_b/nlc_c (ωB97X-V, ωB97M-V) gets the VV10 nonlocal-correlation double-grid term added on the materialized XC grid.

dftax.energy.xc.functional(name: str) -> XCFunctional

The XC functional named name, ready to pass to :class:~dftax.KS.

Lookup ignores case and punctuation, so "wB97X-V", "wb97xv" and "WB97X_V" all resolve to the same functional. Dispersion stays a separate axis (KS(..., dispersion=d3bj())), matching the library's choices-as-values style: a functional is a functional, not a functional plus a correction.

Example

KS(mol, functional("r2scan"))
KS(mol, functional("b3lyp"), dispersion=d3bj())

dftax.energy.xc.LDA

Local density approximation: Slater exchange + VWN5 correlation.

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density: Float[Array, s] | Shaped[Array, ''], *args) -> Shaped[Array, '']

VWN5 LDA correlation (parameterization V from VWN 1980).

exchange(density: Float[Array, s] | Shaped[Array, '']) -> Shaped[Array, '']

Local density approximation (LDA) exchange energy density.

dftax.energy.xc.PBE

Convenience wrapper for the full PBE XC functional.

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

PBE correlation energy density with PW92 base.

exchange(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

PBE GGA exchange energy density.

dftax.energy.xc.PBE0

PBE0 global hybrid: ε_xc = 0.75·ε_x^PBE + ε_c^PBE, plus 25% exact exchange.

The HF exchange fraction (0.25) is exposed as hf_coeff; the exact-K contribution is added by the energy driver (see :mod:dftax.ks.energy).

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

hf_coeff class-attribute

Convert a string or number to a floating-point number, if possible.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__call__(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

PBE correlation energy density with PW92 base.

exchange(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

PBE GGA exchange energy density.

dftax.energy.xc.B3LYP

Gaussian / libxc default B3LYP functional (uses VWN_RPA).

Total XC energy density (per electron): ε_xc = 0.08 · ε_x^LDA + 0.72 · ε_x^B88 + 0.19 · ε_c^VWN_RPA + 0.81 · ε_c^LYP

The HF exchange coefficient (0.20) is exposed as hf_coeff. The EnergyFunctional is responsible for building the K matrix and adding -0.5 · hf_coeff · Tr(P K) to the total energy.

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

ac class-attribute

Convert a string or number to a floating-point number, if possible.

ax class-attribute

Convert a string or number to a floating-point number, if possible.

bc class-attribute

Convert a string or number to a floating-point number, if possible.

bx class-attribute

Convert a string or number to a floating-point number, if possible.

hf_coeff class-attribute

Convert a string or number to a floating-point number, if possible.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__call__(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

LYP GGA correlation energy density.

Closed-shell expression (ρ_↑ = ρ_↓ = ρ/2) from Miehlich et al. (1989):

ε_c^LYP = -a · g(ρ)
         - a·b·ω(ρ) · [ 2^{11/3} C_F · (ρ/2)^{8/3}/ρ
                        + (1/36) · (47 - 7 δ) · γ/ρ
                        - (5/4) · (γ_↑↑ + γ_↓↓) · (1/ρ)   (γ_σσ = |∇ρ_σ|² = γ/4)
                        - ... ]

The exact algebraic form has been re-derived for ρ_↑ = ρ_↓ = ρ/2 and validated against pyscf.dft.libxc.eval_xc(',LYP', ...).

exchange(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

Becke 1988 GGA exchange.

Per spin channel

ε_x^{B88,σ}(ρ_σ, |∇ρ_σ|) = -C_σ ρ_σ^{1/3} - β ρ_σ^{1/3} x_σ^2 / (1 + 6 β x_σ sinh^{-1}(x_σ))

with x_σ = |∇ρ_σ| / ρ_σ^{4/3}, β = 0.0042, C_σ = (3/4)(6/π)^{1/3}.

dftax.energy.xc.CAMB3LYP

CAM-B3LYP (Yanai-Tew-Handy 2004) range-separated hybrid.

DFT part (identified against libxc HYB_GGA_XC_CAM_B3LYP to machine precision): 0.35·B88 + 0.46·SR-B88(ITYH, ω=0.33) + 0.81·LYP + 0.19·VWN5. Exact exchange: 0.19·K + 0.46·K_lr(ω=0.33) (PySCF rsh_coeff convention (0.33, 0.65, -0.46) re-expressed as full-range + long-range).

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

hf_coeff class-attribute

Convert a string or number to a floating-point number, if possible.

hf_coeff_lr class-attribute

Convert a string or number to a floating-point number, if possible.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

omega class-attribute

Convert a string or number to a floating-point number, if possible.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__call__(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

LYP GGA correlation energy density.

Closed-shell expression (ρ_↑ = ρ_↓ = ρ/2) from Miehlich et al. (1989):

ε_c^LYP = -a · g(ρ)
         - a·b·ω(ρ) · [ 2^{11/3} C_F · (ρ/2)^{8/3}/ρ
                        + (1/36) · (47 - 7 δ) · γ/ρ
                        - (5/4) · (γ_↑↑ + γ_↓↓) · (1/ρ)   (γ_σσ = |∇ρ_σ|² = γ/4)
                        - ... ]

The exact algebraic form has been re-derived for ρ_↑ = ρ_↓ = ρ/2 and validated against pyscf.dft.libxc.eval_xc(',LYP', ...).

exchange(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

Becke 1988 GGA exchange.

Per spin channel

ε_x^{B88,σ}(ρ_σ, |∇ρ_σ|) = -C_σ ρ_σ^{1/3} - β ρ_σ^{1/3} x_σ^2 / (1 + 6 β x_σ sinh^{-1}(x_σ))

with x_σ = |∇ρ_σ| / ρ_σ^{4/3}, β = 0.0042, C_σ = (3/4)(6/π)^{1/3}.

dftax.energy.xc.WB97X

ωB97X (Chai-Head-Gordon 2008) range-separated hybrid GGA.

Exchange: B97 series over SR-LDA (ω = 0.3); correlation: B97 series over Stoll-partitioned PW92 (same-spin / opposite-spin). Exact exchange: 0.157706·K + 0.842294·K_lr(ω=0.3). The series coefficients match libxc HYB_GGA_XC_WB97X (recovered by exact linear fit, residual ~1e-10) and the published parameters.

CAB class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

CSS class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

CX class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

hf_coeff class-attribute

Convert a string or number to a floating-point number, if possible.

hf_coeff_lr class-attribute

Convert a string or number to a floating-point number, if possible.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

omega class-attribute

Convert a string or number to a floating-point number, if possible.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__call__(density: Float[Array, s] | Shaped[Array, ''], grad_density: Float[Array, 'i s'] | Float[Array, i]) -> Shaped[Array, '']

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density: Float[Array, s] | Shaped[Array, '']) -> Shaped[Array, '']

Perdew-Wang 1992 LSDA correlation (the B97 family's LSDA backbone).

ε_c(r_s, ζ) = ε_0 - α_c·f(ζ)/f''(0)·(1-ζ⁴) + (ε_1-ε_0)·f(ζ)·ζ⁴ with the three G-function fits of PW92 (the parameters fit −α_c, hence the sign). Validated against libxc LDA_C_PW to ~4e-10 (libxc carries one more digit on the A constants).

exchange(density: Float[Array, s] | Shaped[Array, '']) -> Shaped[Array, '']

Local density approximation (LDA) exchange energy density.

dftax.energy.xc.WB97XV

ωB97X-V (Mardirossian-Head-Gordon 2014): 10-parameter RSH GGA + VV10.

The B97 series is deliberately short (3 exchange, 2 + 2 correlation terms); nonlocal correlation is VV10 with b = 6.0, C = 0.01 (declared via nlc_b/nlc_c; the KS builder adds the double-grid term). Exact exchange 0.167·K + 0.833·K_lr(ω = 0.3). Series coefficients verified pointwise against libxc HYB_GGA_XC_WB97X_V.

CAB class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

CSS class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

CX class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

hf_coeff class-attribute

Convert a string or number to a floating-point number, if possible.

hf_coeff_lr class-attribute

Convert a string or number to a floating-point number, if possible.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

nlc_b class-attribute

Convert a string or number to a floating-point number, if possible.

nlc_c class-attribute

Convert a string or number to a floating-point number, if possible.

omega class-attribute

Convert a string or number to a floating-point number, if possible.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

dftax.energy.xc.WB97MV

ωB97M-V (Mardirossian-Head-Gordon 2016): 12-parameter RSH mGGA + VV10.

The B97 double power series g = Σ c_ij·w^i·u^j over the same bases as ωB97X (SR-LDA exchange at ω = 0.3, Stoll-partitioned PW92 correlation), with the meta ingredient entering through w = (τ^unif − τ)/(τ^unif + τ) per spin (τ^unif_σ = (3/10)(6π²)^{2/3} ρ_σ^{5/3}); the opposite-spin channel uses the p-weighted average of the two channels' w (p = τ/(τ^unif + τ)) and the mean-square gradient variable, exactly the libxc b97mv scheme. Exact exchange 0.15·K + 0.85·K_lr(ω=0.3) (libxc CAM alpha=1, beta=−0.85 re-expressed as full-range + long-range); VV10 with b = 6.0, C = 0.01. Coefficients and term selection verbatim from libxc HYB_MGGA_XC_WB97M_V / the published parameters; validated pointwise against libxc.

COS class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

CSS class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

CX class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

hf_coeff class-attribute

Convert a string or number to a floating-point number, if possible.

hf_coeff_lr class-attribute

Convert a string or number to a floating-point number, if possible.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

nlc_b class-attribute

Convert a string or number to a floating-point number, if possible.

nlc_c class-attribute

Convert a string or number to a floating-point number, if possible.

omega class-attribute

Convert a string or number to a floating-point number, if possible.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__call__(density, grad_density, tau) -> Shaped[Array, '']

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density: Float[Array, s] | Shaped[Array, '']) -> Shaped[Array, '']

Perdew-Wang 1992 LSDA correlation (the B97 family's LSDA backbone).

ε_c(r_s, ζ) = ε_0 - α_c·f(ζ)/f''(0)·(1-ζ⁴) + (ε_1-ε_0)·f(ζ)·ζ⁴ with the three G-function fits of PW92 (the parameters fit −α_c, hence the sign). Validated against libxc LDA_C_PW to ~4e-10 (libxc carries one more digit on the A constants).

exchange(density: Float[Array, s] | Shaped[Array, '']) -> Shaped[Array, '']

Local density approximation (LDA) exchange energy density.

dftax.energy.xc.R2SCAN

r2SCAN meta-GGA (regularized-restored SCAN; the modern default mGGA).

Needs the kinetic-energy density τ on the grid (xc_type = "MGGA"); the KS terms provide it and the Fock matrix comes from autodiff of the energy, so no vxc code exists anywhere. Validated pointwise against libxc mgga_x_r2scan/mgga_c_r2scan to machine precision.

__abstractclassvars__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractmethods__ class-attribute

Build an immutable unordered collection of unique elements.

__abstractvars__ class-attribute

Build an immutable unordered collection of unique elements.

__annotations__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_fields__ class-attribute

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object's (key, value) pairs dict(iterable) -> new dictionary initialized as if via: d = {} for k, v in iterable: d[k] = v dict(**kwargs) -> new dictionary initialized with the name=value pairs in the keyword argument list. For example: dict(one=1, two=2)

__dataclass_params__ class-attribute

__doc__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__firstlineno__ class-attribute

int([x]) -> integer int(x, base=10) -> integer

Convert a number or string to an integer, or return 0 if no arguments are given. If x is a number, return x.int(). For floating-point numbers, this truncates towards zero.

If x is not a number or if base is given, then x must be a string, bytes, or bytearray instance representing an integer literal in the given base. The literal can be preceded by '+' or '-' and be surrounded by whitespace. The base defaults to 10. Valid bases are 0 and 2-36. Base 0 means to interpret the base from the string as an integer iteral.

int('0b100', base=0) 4

__match_args__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

__module__ class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__static_attributes__ class-attribute

Built-in immutable sequence.

If no argument is given, the constructor returns an empty tuple. If iterable is specified the tuple is initialized from iterable's items.

If the argument is a tuple, the return value is the same object.

name class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

xc_type class-attribute

str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to 'utf-8'. errors defaults to 'strict'.

__call__(density, grad_density, tau) -> Shaped[Array, '']

__init__() -> None

Initialize self. See help(type(self)) for accurate signature.

correlation(density, grad_density, tau) -> Shaped[Array, '']

r2SCAN correlation (built on the modified-PW92 LSDA).

exchange(density, grad_density, tau) -> Shaped[Array, '']

r2SCAN exchange (spin-scaled per channel).