Basis set information

The optimal choice of the basis set parameters is determined by the problem to be solved.

Therefore, convergence with respect to different parameters was extensively studied for different laser pulses and internuclear separations before the method has been applied for the investigation of the intense-field response of H₂. This section discusses four basis sets which were found to be reasonable for the investigations described in the present chapter.

For small internuclear separations (R <3.0a₀) two different basis sets are chosen, basis A and basis B. Basis A is supposed to be used for shorter wavelengths or pulse durations due to a smaller box size. Basis B is obtained using a larger box size but its application is more expensive.

The basis states ψn,Ω were obtained for every symmetry in basis A as follows. A box size (defined implicitly byξ_maxthat depends onR) of about 350a₀is chosen for different R. Along the ξ coordinate 350 B splines of order k = 10 with an almost linear knot sequence (the 40 first intervals are increased with geometric progression usingg= 1.05, all following intervals have the length of 40th one) were used. Along the η coordinate 30 B splines of order 8 were used in the complete interval−1 ≤η ≤+1, but using the symmetry of a homonuclear system as is described in chapter 2. Out of the resulting 5235 orbitals for every symmetry only 3490 orbitals were further used to construct con-figurations, whereas those orbitals with highly oscillating angular part (with more than 19 nodes for the η-dependent component) were omitted. In most of the subsequent CI calculations approximately 6000 configurations were used for every symmetry. These states result from very long configuration series (3490 configurations) in which one elec-tron occupies the H⁺₂ ground-state 1σ_g orbital while the other one is occupying one of

1 1.5 2

Figure 7.1.: Electronic potential curves of some low-lying states of H2. The the CI results (blue circles) are compared to the quasi-exact values (solid curves). The dashed line shows the ionization threshold of H₂. The different adiabatic electronic states of a given molecular symmetry are numbered in the order of their energies. (Published in [35].)

the remaining, e. g., n πu or n δg orbitals. The other configurations represent doubly-excited situations and are responsible for describing correlation (and real doubly-doubly-excited states). Finally, out of the obtained CI states only those with an energy below the energy cut-off (here chosen to be 10 a.u. above the ionization threshold) were included in the time propagation (about 5400 states per symmetry).

The preparation of the basis states ψ_n,Ω in basis B differs mainly by using a larger box size, which was chosen to be twice as larger as for basis A (about 700a₀). Correspond-ingly, the number of B splines along the ξ coordinate was increased to 500. To reduce the number of orbitals, the number ofB splines along theη coordinate was reduced from 30 to 24. Out of the resulting 5988 orbitals for every symmetry only 4990 orbitals were further used to construct configurations, whereas those orbitals with highly oscillating angular part (with more than 19 nodes for theη-dependent component) were omitted.

The adopted CI configurations differ from those of basis A only by extending the number of configurations in the leading configuration series (where one electron occupies the 1σ_g orbital) from 3490 to 4990. Thus, for every symmetry the full CI configuration series of basis B is 1500 configurations longer than the one of basis A. Applying the same cut-off results in about 6900 states per symmetry included in the time propagation.

For internuclear separations R > 3a₀ the probability to find H⁺₂ in the excited 1σ_u state after ionization process becomes significant. Numerical tests have shown that the discussed basis sets have to be extended in order to obtain converged photoelectron energy spectra. For this purpose, all configurations in which one electron occupies the H⁺₂ 1σu orbital were additionally included in the configuration series. This resulted in an almost by a factor of 2 larger number of states per symmetry included in the time propagation. These basis sets are labeled C (extended basis set A) and D (extended basis set B).

A variation of the remaining parameter Nmax^ph (or Λ_max) provides an additional conver-gency test. This parameter can be chosen depending on the problem to be solved and will be specified later.

The CI calculations using all basis sets discussed above yield almost the same electronic energiesEn,Ω for the low-lying states of H₂. Conversely, highly energetic Rydberg states and discretized continuum states have different energies due to a larger box size for basis sets B and D as compared to basis sets A and C. Figure 7.1 shows the electronic ener-gies En,Ω of various low-lying states obtained using the discussed basis sets (compared with quasi-exact energies calculated using state-optimized basis set. The agreement is excellent for all states except for 1¹Σ⁺_g and 1¹Σ⁺_u (at larger internuclear distances) where the electronic motion is highly correlated and cannot efficiently be described by a CI calculation employing orbitals with no electron-electron interaction included. Neverthe-less, even for these two states the obtained electronic energies are much better than those obtained with the Hartree-Fock approximation. For example, for the ground state of H₂ with the exact electronic energy at R = 1.4a₀ being equal to −1.1745 a.u., the Hartree-Fock limit is−1.1336 a.u., whereas the present CI calculation yields−1.1604 a.u.

For the subsequent discussion it is helpful to keep in mind the relevant energies and transition frequencies (or wavelengths) of a number of electronic bound states of H₂ that can resonantly be excited by a laser with the corresponding photon frequency.

Since the exact positions of the resonances depend on the adopted electronic structure model, Table 7.1 reports the energies obtained with the present approach and basis set A.

Furthermore, Table 7.1 provides the ground-state energy of H⁺₂ which allows to calculate the exact position of the differentN-photon thresholds. (The 1-photon threshold is given explicitly.)

Table 7.1.: Electronic energies E (in a. u.) of various H₂ states as they are obtained with the basis set A used in this work and the resulting resonantN-photon transition frequencies ω (in eV) and wavelengths λ (in nm). The last row shows the ground-state energy of H⁺₂ and the corresponding 1-photon ionization threshold. (Published in [34].)

State E^a, a.u. E^b, a.u. N ω^a, eV ω^b, eV λ^a, nm λ^b, nm 1¹Σ⁺_g (X) -1.160351 -1.128787

2¹Σ⁺g (EF) -0.690087 -0.716303 2 6.3982 5.6121 193.778 220.922

4 3.1991 2.8060 387.556 441.844

3¹Σ⁺g (GK) -0.626453 -0.660305 2 7.2640 6.3739 170.682 194.515

4 3.6320 3.1870 341.364 389.030

1¹Σ⁺u (B) -0.702364 -0.745749 1 12.4623 10.4229 99.486 118.953

3 4.1541 3.4743 298.458 356.859

2¹Σ⁺u (B’) -0.627569 -0.663476 1 14.4975 12.6616 85.520 97.920 3¹Σ⁺u (B”) -0.602079 -0.636115 1 15.1914 13.4063 81.614 92.481

1¹Πu(C) -0.687338 -0.716903 1 12.8712 11.2078 96.326 110.622

3 4.2904 3.7359 288.978 331.866

2¹Πu(D) -0.623117 -0.654839 1 14.6187 12.8966 84.811 96.136

1¹∆g (J) -0.625213 -0.657517 2 7.2808 6.4119 170.286 193.364

2¹∆g (S) -0.601098 -0.633603 2 7.6089 6.7372 162.943 184.026

1σg [H⁺₂] -0.569984 -0.602634 1 16.0645 14.3171 77.178 86.597

a For the internuclear distanceR= 1.4a0 b

For the internuclear distanceR= 2.0a0