multidip_asy Module Reference

Asymptotic approximation of multi-photon matrix elements. More...

Functions/Subroutines
real(wp) function	M1_H1s (l, k)
	Hydrogen 1s one-photon ionization.
complex(wp) function	Icca (z, a, b, k, l, kappa, lambda)
	Near-field part of the continuum-continuum transition integral.
complex(wp) function	I2cca (z, a, kf, lf, kn, ln, ki, li)
	Near-field part of the double continuum-continuum transition integral.
complex(wp) function	Acca (z, a, b, k, l, kappa, lambda)
	Far-field part of the continuum-continuum transition integral.
complex(wp) function	A2cca (z, a, kf, lf, kn, ln, ki, li)
	Far-field part of the double continuum-continuum transition integral.
complex(wp) function	Icc (z, b, k, l, kappa, lambda)
	Continuum-continuum transition integral.
complex(wp) function	I2cc (z, kf, lf, kn, ln, ki, li)
	Two-dimensional continuum-continuum transition integral.
complex(wp) function	Akkl (kappa, lambda, k, l, b)
	Partial-wave-dependent asymptotic correction.
complex(wp) function	Akkl3 (ki, li, kn, ln, kf, lf)
	Third-order asymptotic correction.
complex(wp) function	Akk (kappa, k, l, cc)
	Prefactor in asymptotic 2-photon matrix element.
complex(wp) function	M2_H1s (b, ln, kn, lf, kf)
	Two-photon ionization of hydrogen atom.

Detailed Description

Asymptotic approximation of multi-photon matrix elements.

Author

J Benda

Date

2024

Function/Subroutine Documentation

A2cca()

complex(wp) function multidip_asy::A2cca	(	real(wp), intent(in)	z,
		real(wp), intent(in)	a,
		real(wp), intent(in)	kf,
		integer, intent(in)	lf,
		real(wp), intent(in)	kn,
		integer, intent(in)	ln,
		real(wp), intent(in)	ki,
		integer, intent(in)	li )

Far-field part of the double continuum-continuum transition integral.

Authors

J Benda

Date

2024

Asymptotically evaluates the integral

\[ A_{cc}(a) = \int\limits_a^{+\infty} \int\limits_a^{+\infty} F_{l_f}(k_f,r) r g_{l_n}^{(+)}(k_n; r, r') r' H_{l_i}^+(k_i,r') dr dr' \,. \]

Definition at line 288 of file multidip_asy.f90.

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Acca()

complex(wp) function multidip_asy::Acca	(	real(wp), intent(in)	z,
		real(wp), intent(in)	a,
		integer, intent(in)	b,
		real(wp), intent(in)	k,
		integer, intent(in)	l,
		real(wp), intent(in)	kappa,
		integer, intent(in)	lambda )

Far-field part of the continuum-continuum transition integral.

Authors

J Benda

Date

2024

Asymptotically evaluates the integral

\[ A_{cc}(a) = \int\limits_a^{+\infty} F_l(k,r) r^b H_\lambda^+(\kappa,r) dr \,. \]

Definition at line 262 of file multidip_asy.f90.

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Akk()

complex(wp) function multidip_asy::Akk	(	real(wp), intent(in)	kappa,
		real(wp), intent(in)	k,
		integer, intent(in)	l,
		integer, intent(in)	cc )

Prefactor in asymptotic 2-photon matrix element.

Author

J Benda

Date

2021 - 2024

Target-independent complex prefactor in the 2-photon ionization matrix element as obtained in the asymptotic theory of Dahlström et al (2013), including the long-range amplitude correction.

Definition at line 468 of file multidip_asy.f90.

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Akkl()

complex(wp) function multidip_asy::Akkl	(	real(wp), intent(in)	kappa,
		integer, intent(in)	lambda,
		real(wp), intent(in)	k,
		integer, intent(in)	l,
		integer, intent(in)	b )

Partial-wave-dependent asymptotic correction.

Authors

J Benda

Date

2024

Approximate correcting factor for calculation of two-photon matrix element from a one-photon matrix element.

Definition at line 425 of file multidip_asy.f90.

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Akkl3()

complex(wp) function multidip_asy::Akkl3	(	real(wp), intent(in)	ki,
		integer, intent(in)	li,
		real(wp), intent(in)	kn,
		integer, intent(in)	ln,
		real(wp), intent(in)	kf,
		integer, intent(in)	lf )

Third-order asymptotic correction.

Authors

J Benda

Date

2024

Definition at line 445 of file multidip_asy.f90.

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I2cc()

complex(wp) function multidip_asy::I2cc	(	real(wp), intent(in)	z,
		real(wp), intent(in)	kf,
		integer, intent(in)	lf,
		real(wp), intent(in)	kn,
		integer, intent(in)	ln,
		real(wp), intent(in)	ki,
		integer, intent(in)	li )

Two-dimensional continuum-continuum transition integral.

Authors

J Benda

Date

2024

Calculate the integral

\[ I_{cc} = \int\limits_0^{+\infty}\int\limits_0^{+\infty} F_{l_f}(k_f,r) r g_{l_n}^{(+)}(k_n;r,r') r' H_{l_i}^+(k_i,r') dr dr' = I_1 + I_2 + I_3 + I_4 \]

by separation into four contributions. The near-field region ( \( r < a, r' < a \)) is integrated numerically,

\[ I_1 = \int\limits_0^a \int\limits_0^a F_{l_f}(k_f,r) r g_{l_n}^{(+)}(k_n;r,r') r' H_{l_i}^+(k_i,r') dr dr' \,, \]

the far-field region is integrated asymptotically,

\[ I_2 = \int\limits_a^{+\infty} \int\limits_a^{+\infty} F_{l_f}(k_f,r)rg_{l_n}^{(+)}(k_n;r,r')r'H_{l_i}^+(k_i,r') drdr'\,, \]

and the mixed regions factorize into one-dimensional integrals

\[ I_3 = -\frac{2}{k_n} \int\limits_a^{+\infty} F_{l_f}(k_f,r) r H_{l_n}^+(k_n,r) dr \int\limits_0^a F_{l_n}(k_n,r') r' H_{l_i}^+(k_n,r') dr' \,, \]

\[ I_4 = -\frac{2}{k_n} \int\limits_0^a F_{l_f}(k_f,r) r F_{l_n}(k_n,r) dr \int\limits_0^{+\infty} H_{l_n}^+(k_n,r') r' H_{l_i}^+(k_n,r') dr' \,. \]

Definition at line 361 of file multidip_asy.f90.

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I2cca()

complex(wp) function multidip_asy::I2cca	(	real(wp), intent(in)	z,
		real(wp), intent(in)	a,
		real(wp), intent(in)	kf,
		integer, intent(in)	lf,
		real(wp), intent(in)	kn,
		integer, intent(in)	ln,
		real(wp), intent(in)	ki,
		integer, intent(in)	li )

Near-field part of the double continuum-continuum transition integral.

Authors

J Benda

Date

2024

Numerically evaluate the integral

\[ I = \int\limits_0^a \int\limits_0^a F_{l_f}(k_f,r) r g_{l_n}^{(+)}(k_n; r, r') r' H_{l_i}^+(r') dr' \,. \]

Definition at line 176 of file multidip_asy.f90.

Icc()

complex(wp) function multidip_asy::Icc	(	real(wp), intent(in)	z,
		integer, intent(in)	b,
		real(wp), intent(in)	k,
		integer, intent(in)	l,
		real(wp), intent(in)	kappa,
		integer, intent(in)	lambda )

Continuum-continuum transition integral.

Authors

J Benda

Date

2024

Calculate integral of regular Coulomb function times r^b times the outgoing Coulomb-Hankel function:

\[ I_{cc} = \int\limits_0^{+\infty} F_l(k,r) r^b H_\lambda^+(\kappa,r) dr \,. \]

Definition at line 320 of file multidip_asy.f90.

Icca()

complex(wp) function multidip_asy::Icca	(	real(wp), intent(in)	z,
		real(wp), intent(in)	a,
		integer, intent(in)	b,
		real(wp), intent(in)	k,
		integer, intent(in)	l,
		real(wp), intent(in)	kappa,
		integer, intent(in)	lambda )

Near-field part of the continuum-continuum transition integral.

Authors

J Benda

Date

2024

Numerically evaluates the integral

\[ I_{cc}(a) = \int\limits_0^a F_l(k,r) r^b H_\lambda^+(\kappa,r) dr \,. \]

This function splits the integral to two pieces at the classical turning point \( a_0 \) of the Coulomb functions. The non-oscillatory, classically forbidden part \( (0, a_0) \) is integrated using Romberg quadrature, while the remainder \( (a_0, a) \) is integrated using Levin quadrature.

Definition at line 74 of file multidip_asy.f90.

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M1_H1s()

real(wp) function multidip_asy::M1_H1s	(	integer, intent(in)	l,
		real(wp), intent(in)	k )

Hydrogen 1s one-photon ionization.

Author

J Benda

Date

2024

Evaluate the radial factor in the one-photon ionization amplitude of H(1s). This can be expressed as

\[ M^{(1)} = \int_0^\infty F_l(k, r) r P_{10}(r) dr \,. \]

Definition at line 43 of file multidip_asy.f90.

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M2_H1s()

complex(wp) function multidip_asy::M2_H1s	(	integer, intent(in)	b,
		integer, intent(in)	ln,
		real(wp), intent(in)	kn,
		integer, intent(in)	lf,
		real(wp), intent(in)	kf )

Two-photon ionization of hydrogen atom.

Authors

J Benda

Date

2024

Evaluate the full partial amplitude of two-photon ionization of hydrogen atom from the ground state via the s→p→lf angular momentum pathway. The result is proportional to a double indefinite integral, but one of the integrations is effectively definite due to the exponential decay of the hydrogen bound state wave function. Disregarding some normalization factors we have

\[ M_l^{(2)} = i^{-l} e^{i\sigma_l} \int_0^\infty \int_0^\infty F_l(r) r^b F_1(r_<) H_1^+(r_>) r' P_{10}(r') dr dr'. \]

The integral can be decomposed into two contributions, the finite-range double integral

\[ I = \int_0^a \int_0^a F_l(r) r^b F_1(r_<) H_1^+(r_>) r' P_{10}(r') dr dr', \]

which is integrated by means of a two-dimensional Romberg quadrature, and the product of two one-dimensional integrals

\[ J = \int_a^\infty F_l(r) r^b H_1^+(r) dr \int_0^\infty F_1(r) r P_{10}(r) dr , \]

of which the first is integrated by asymptotic integration routine from MULTIDIP and the second has a closed form. The radius "a" has to be sufficiently large so that the electron density of the bound state vanishes and also to make sure that the asymptotic integration gives meaningful results. In the present implementation it is set to 100 atomic units.

Definition at line 530 of file multidip_asy.f90.

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