Time-dependent electron localization function
[Phys. Rev. A (Rap. Comm.) 71, 10501 (2005)]

Content

• 0 Overview
• 1 Excitation of ethyne (actylene) by a laser
     — ν = 17.15 eV (E₀: 3 eV, 0.5 eV), ν = 13.35 eV (E₀: 3 eV, 0.5 eV), ν = 9.55 eV (E₀: 3 eV, 0.5 eV)
• 2 Scattering of a proton by ethene (ethylene)
     — against a carbon · against a hydrogen · through the π bond [with 301 keV, 2 keV, 482 eV, 120 eV]
• 3 Excitation of nitrogen (N₂) by a laser
     — ν = 12.45 eV (E₀: 5 eV, 2 eV), ν = 13.65 eV (E₀: 5 eV, 2 eV)
• 4 References

0 Overview

All ELF images shown here have either a isosurface or a contour line at ELF = 0.8 and use this colour scheme. For the density the same colour scheme has been applied, with the exception that n(r, t) = 0.4 equals the highest value (white), values that are higher are shown in black. In the logarithmic density plot, the same colour scheme as for the ELF has been used, i.e. the shown density is calculated as n'(r,  t) = log (1 + n(r, t))/log 2.

Note that the bonds are drawn based only on the distance between the atoms and are therefore not always physically reasonable.

File sizes: The .png files are between 3 kB and 9 kB, the .mpeg files between 2 MB and 18 MB.

1 Excitation of ethyne (actylene) by a laser

1.1 E₀ = 3 eV/Å with ν = 17.15 eV

The laser has a maximal intensity of I₀ = 1.19×10¹⁴ W·cm^-2 (E₀ = 3 eV/Å) and a frequency of ν = 17.15 eV = 4146 THz ⇔ λ = 72.3 nm = 723 Å.

• FIG 1.1a: ELF contour (mpeg) — FIG 1.1b: ELF iso (mpeg) — FIG 1.1c: ELF iso (without slab) (mpeg)
• FIG 1.1d: Electronic density (logscale) (mpeg) — FIG 1.1e: Electronic density (mpeg)
• FIG 1.1f: Dipol moment (png)
• FIG 1.1g: Total energy (png)
• FIG 1.1h: Laser intensity over time (png)

1.2 E₀ = 0.5 eV/Å with ν = 17.15 eV

The laser has a maximal intensity of I₀ = 3.318×10¹³ W·cm^-2 (E₀ = 0.5 eV/Å) and a frequency of ν = 17.15 eV = 4146 THz ⇔ λ = 72.3 nm = 723 Å.

• FIG 1.2a: ELF contour (mpeg) — FIG 1.2b: ELF iso (mpeg) — FIG 1.2c: ELF iso (without slab) (mpeg)
• FIG 1.2d: Electronic density (logscale) (mpeg) — FIG 1.2e: Electronic density (mpeg)
• FIG 1.2f: Dipol moment (png)
• FIG 1.2g: Total energy (png)
• FIG 1.2h: Laser intensity over time (png)

1.3 E₀ = 3 eV/Å with ν = 13.35 eV

The laser has a maximal intensity of I₀ = 1.19×10¹⁴ W·cm^-2 (E₀ = 3 eV/Å) and a frequency of ν = 13.35 eV = 3010 THz ⇔ λ = 99.6 nm = 996 Å.

• FIG 1.3a: ELF contour (mpeg) — FIG 1.3b: ELF iso (mpeg) — FIG 1.3c: ELF iso (without slab) (mpeg)
• FIG 1.3d: Electronic density (logscale) (mpeg) — FIG 1.3e: Electronic density (mpeg)
• FIG 1.3f: Dipol moment (png)
• FIG 1.3g: Total energy (png)
• FIG 1.3h: Laser intensity over time (png)

1.4 E₀ = 0.5 eV/Å with ν = 13.35 eV

The laser has a maximal intensity of I₀ = 3.318×10¹³ W·cm^-2 (E₀ = 0.5 eV/Å) and a frequency of ν = 13.35 eV = 3010 THz ⇔ λ = 99.6 nm = 996 Å.

• FIG 1.4a: ELF contour (mpeg) — FIG 1.4b: ELF iso (mpeg) — FIG 1.4c: ELF iso (without slab) (mpeg)
• FIG 1.4d: Electronic density (logscale) (mpeg) — FIG 1.4e: Electronic density (mpeg)
• FIG 1.4f: Dipol moment (png)
• FIG 1.4g: Total energy (png)
• FIG 1.4h: Laser intensity over time (png)

1.5 E₀ = 3 eV/Å with ν = 9.55 eV

The laser has a maximal intensity of I₀ = 1.19×10¹⁴ W·cm^-2 (E₀ = 3 eV/Å) and a frequency of ν = 9.55 eV = 2309 THz ⇔ λ = 129.8 nm = 1298 Å.

• FIG 1.5a: ELF contour (mpeg) — FIG 1.5b: ELF iso (mpeg) — FIG 1.5c: ELF iso (without slab) (mpeg)
• FIG 1.5d: Electronic density (logscale) (mpeg) — FIG 1.5e: Electronic density (mpeg)
• FIG 1.5f: Dipol moment (png)
• FIG 1.5g: Total energy (png)
• FIG 1.5h: Laser intensity over time (png)

1.6 E₀ = 0.5 eV/Å with ν = 9.55 eV

The laser has a maximal intensity of I₀ = 3.318×10¹³ W·cm^-2 (E₀ = 0.5 eV/Å) and a frequency of ν = 9.55 eV = 2309 THz ⇔ λ = 129.8 nm = 1298 Å.

• FIG 1.6a: ELF contour (mpeg) — FIG 1.6b: ELF iso (mpeg) — FIG 1.6c: ELF iso (without slab) (mpeg)
• FIG 1.6d: Electronic density (logscale) (mpeg) — FIG 1.6e: Electronic density (mpeg)
• FIG 1.6f: Dipol moment (png)
• FIG 1.6g: Total energy (png)
• FIG 1.6h: Laser intensity over time (png)

2 Scattering of a proton by ethene (ethylene)

Time-dependent ELF for the scattering of a fast, non-relativistic proton by ethene.

2.1 Proton against a carbon

The proton has the velocity v = 1.02×10⁵ m/s (kinetic energy E_kin = 2 keV).

• FIG 2.1a: ELF contour (mpeg) — FIG 2.1b: ELF iso (mpeg) — FIG 2.1c: ELF iso (without slab) (mpeg)
• FIG 2.1d: Electronic density (logscale) (mpeg) — FIG 2.1e: Electronic density (mpeg)
• FIG 2.1f: Number of electrons (inside the box) (png)
• FIG 2.1g: Total energy (png) — FIG 2.1h: Kinetic energy (png) — FIG 2.1i: Electronic energy (png)

2.2 Proton against a hydrogen

The proton has the velocity v = 3.8×10⁵ m/s (kinetic energy E_kin = 3 keV).

• FIG 2.2a: ELF contour (mpeg) — FIG 2.2b: ELF iso (mpeg) — FIG 2.2c: ELF iso (without slab) (mpeg)
• FIG 2.2d: Electronic density (logscale) (mpeg) — FIG 2.2e: Electronic density (mpeg)
• FIG 2.2f: Number of electrons (inside the box) (png)
• FIG 2.2g: Total energy (png) — FIG 2.2h: Kinetic energy (png) — FIG 2.2i: Electronic energy (png)

2.2 Proton through the π bond

301 keV

The proton has the velocity v = 7.6×10⁶ m/s (kinetic energy E_kin = 301 keV).

• FIG 2.3a: ELF contour (mpeg) — FIG 2.3b: ELF iso (mpeg) — FIG 2.3c: ELF iso (without slab) (mpeg)
• FIG 2.3d: Electronic density (logscale) (mpeg) — FIG 2.3e: Electronic density (mpeg)
• FIG 2.3f: Number of electrons (inside the box) (png)
• FIG 2.3g: Total energy (png) — FIG 2.3h: Kinetic energy (png, 3ng) — FIG 2.3i: Electronic energy (png)

2 keV

The proton has the velocity v = 6.1×10⁵ m/s (kinetic energy E_kin = 2 keV).

• FIG 2.4a: ELF contour (mpeg) — FIG 2.4b: ELF iso (mpeg) — FIG 2.4c: ELF iso (without slab) (mpeg)
• FIG 2.4d: Electronic density (logscale) (mpeg) — FIG 2.4e: Electronic density (mpeg)
• FIG 2.4f: Number of electrons (inside the box) (png)
• FIG 2.4g: Total energy (png) — FIG 2.4h: Kinetic energy (png) — FIG 2.4i: Electronic energy (png)

482 eV

The proton has the velocity v = 3.0×10⁵ m/s (kinetic energy E_kin = 482 keV).

• FIG 2.5a: ELF contour (mpeg) — FIG 2.5b: ELF iso (mpeg) — FIG 2.5c: ELF iso (without slab) (mpeg)
• FIG 2.5d: Electronic density (logscale) (mpeg) — FIG 2.5e: Electronic density (mpeg)
• FIG 2.5f: Number of electrons (inside the box) (png)
• FIG 2.5g: Total energy (png) — FIG 2.5h: Kinetic energy (png) — FIG 2.5i: Electronic energy (png)

120 eV

The proton has the velocity v = 1.5×10⁵ m/s (kinetic energy E_kin = 120 keV).

• FIG 2.6a: ELF contour (mpeg) — FIG 2.6b: ELF iso (mpeg) — FIG 2.6c: ELF iso (without slab) (mpeg)
• FIG 2.6d: Electronic density (logscale) (mpeg) — FIG 2.6e: Electronic density (mpeg)
• FIG 2.6f: Number of electrons (inside the box) (png)
• FIG 2.6g: Total energy (png) — FIG 2.6h: Kinetic energy (png) — FIG 2.6i: Electronic energy (png)

3 Excitation of nitrogen (N₂) by a laser

3.1 E₀ = 5 eV/Å with ν = 12.45 eV

The laser has a maximal intensity of I₀ = 3.318×10¹⁴ W·cm^-2 (E₀ = 5 eV/Å) and a frequency of ν = 12.45 eV = 3010 THz ⇔ λ = 99.6 nm = 996 Å.

• FIG 3.1a: ELF contour (mpeg) — FIG 3.1b: ELF iso (mpeg) — FIG 3.1c: ELF iso (without slab) (mpeg)
• FIG 3.1d: Electronic density (logscale) (mpeg) — FIG 3.1e: Electronic density (mpeg)
• FIG 3.1f: Dipol moment (png)
• FIG 3.1g: Total energy (png)
• FIG 3.1h: Laser intensity over time (png)

3.2 E₀ = 2 eV/Å with ν = 12.45 eV

The laser has a maximal intensity of I₀ = 5.309×10¹³ W·cm^-2 (E₀ = 2 eV/Å) and a frequency of ν = 12.45 eV = 3010 THz ⇔ λ = 99.6 nm = 996 Å.

• FIG 3.2a: ELF contour (mpeg) — FIG 3.2b: ELF iso (mpeg) — FIG 3.2c: ELF iso (without slab) (mpeg)
• FIG 3.2d: Electronic density (logscale) (mpeg) — FIG 3.1e: Electronic density (mpeg)
• FIG 3.2f: Dipol moment (png)
• FIG 3.2g: Total energy (png)
• FIG 3.2h: Laser intensity over time (png)

3.3 E₀ = 5 eV/Å with ν = 13.65 eV

The laser has a maximal intensity of I₀ = 3.318×10¹⁴ W·cm^-2 (E₀ = 5 eV/Å) and a frequency of ν = 13.65 eV = 3301 THz ⇔ λ = 90.8 nm = 908 Å.

• FIG 3.3a: ELF contour (mpeg) — FIG 3.3b: ELF iso (mpeg) — FIG 3.3c: ELF iso (without slab) (mpeg)
• FIG 3.3d: Electronic density (logscale) (mpeg) — FIG 3.3e: Electronic density (mpeg)
• FIG 3.3f: Dipol moment (png)
• FIG 3.3g: Total energy (png)
• FIG 3.3h: Laser intensity over time (png)

3.4 E₀ = 2 eV/Å with ν = 13.65 eV

The laser has a maximal intensity of I₀ = 5.309×10¹³ W·cm^-2 (E₀ = 2 eV/Å) and a frequency of ν = 12.45 eV = 3301 THz ⇔ λ = 90.8 nm = 908 Å.

• FIG 3.4a: ELF contour (mpeg) — FIG 3.4b: ELF iso (mpeg) — FIG 3.4c: ELF iso (without slab) (mpeg)
• FIG 3.4d: Electronic density (logscale) (mpeg) — FIG 3.4e: Electronic density (mpeg)
• FIG 3.4f: Dipol moment (png)
• FIG 3.4g: Total energy (png)
• FIG 3.4h: Laser intensity over time (png)

4 References

• [1] A. D. Becke and K. E. Edgecombe, A simple measure of electron localization in atomic and molecular systems, Journal of Chemical Physics 92, 5397 (1990), doi: 10.1063/1.458517
• [2] Z. Chang et al., Generation of coherent soft X rays at 2.7 nm using high harmonics, Physical Review Letters 79, 2967 (1997), doi: 10.1103/PhysRevLett.79.2967
• [3] Ch. Spielmann et al., Generation of coherent X-rays in the water window using 5-femtosecond laser pulses, Science 278, 661-664 (1997).
• [4] M. Drescher et al., Time-resolved atomic inner-shell spectroscopy, Nature 419 803-807 (2002), doi: 10.1038/nature01143
• [5] D. B. Chesnut, An electron localization function study of the lone pair, Journal of Physical Chemistry A 104, 11644 (2000), doi: 10.1021/jp002957u
• [6] Andreas Savin et al., ELF: The electron Localization Function, Angewandte Chemie Int. Ed. 36, 1808-1823 (1997).
• [7] Jeremy K. Burdett and Thomas A. McCormick, Electron localization in molecules and solids: The meaning of ELF, Journal of Physical Chemistry A 102, 6366-6372 (1998), doi: 10.1021/jp9820774
• [8] Vladimir Tsirelson and Adam Stash, Determination of the electron localization function from electron density, Chemical Physics Letters 351, 142-148 (2002), doi: 10.1016/S0009-2614(01)01361-6
• [9] B. Silvi and A. Savin, Classification of chemical bonds based on topological analysis of electron localization functions, Nature 371, 683-686 (1994).
• [10] http://www.physik.fu-berlin.de/~ag-gross/
• [11] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984), doi: 10.1103/PhysRevLett.52.997
• [12] M. A. L. Marques and E. K. U. Gross, Time-dependent density functional theory, in A Primer in Density Functional Theory, edited by Carlos Fiolhais, Fernando Nogueira, and Miguel A. L. Marques (Springer-Verlag, Berlin, 2003), p. 144. ISBN: 3-540-03083-2
• [13] Miguel A. L. Marques et al., octopus: a first-principles tool for excited electronion dynamics, Computer Physics Communications 151, 60-78 (2002), doi: 10.1016/S0010-4655(02)00686-0