4.2 Multi-Dimensional Molecules
4.2.2 Double Ionization
MOcore
−715.00eV HOMO-2
−43.22eV HOMO-1
−20.07eV HOMOB2
−17.16eV HOMOB1
−17.16eV
0%
5%
10%
15%
20%
25%
30%
D=30 D=30 D=30 D=30 D=30
D=50 D=50 D=50 D=50 D=50
D=70 D=70 D=70 D=70 D=70
D=90 D=90 D=90 D=90 D=90
D=110 D=110 D=110 D=110 D=110
Error
hydrogen fluoride HF
MOcore
−559.29eV HOMO-3
−36.80eV HOMO-2
−14.92eV HOMO-1
−13.55eV HOMO
−13.55eV
D=30 D=30 D=30 D=30 D=30
D=50 D=50 D=50 D=50 D=50
D=70 D=70 D=70 D=70 D=70
D=90 D=90 D=90 D=90 D=90
D=110 D=110 D=110 D=110 D=110
water H2O
MOcore
−422.26eV
HOMO-2
−30.77eV
HOMO-1A0
−17.45eV HOMO-1A00
−17.45eV HOMO
−10.64eV
0%
5%
10%
15%
20%
25%
30%
D=30 D=30 D=30 D=30 D=30
D=50 D=50 D=50 D=50 D=50
D=70 D=70 D=70 D=70 D=70
D=90 D=90 D=90 D=90 D=90
D=110 D=110 D=110 D=110 D=110
Error
ammonia NH3
MOcore
−304.87eV HOMO-1
−25.83eV HOMOA1
−14.83eV HOMOB1
−14.83eV HOMOB2
−14.83eV
D=30 D=30 D=30 D=30 D=30
D=50 D=50 D=50 D=50 D=50
D=70 D=70 D=70 D=70 D=70
D=90 D=90 D=90 D=90 D=90
D=110 D=110 D=110 D=110 D=110
methane CH4
a) b)
c) d)
Figure 4.11.: Relative errors of the OBRDM att = 1f scalculated using the MPS representa-tion for various molecules (hydrogen fluoride, water, ammonia and methane), initial states ( double ionization in HOMO, HOMO-1, HOMO-2, HOMO-3 and MOcore) and bond dimensions (D= 30,D= 50,D= 70,D= 90, andD= 110).
In all calculations (using the complete many-body representation and using the MPS representation) the state is propagated using the orthogonalized Krylov space method with Krylov space dimensionNKry = 6and time step size
∆t = 1.0as. The error of the OBRDM is averaged over a period of±50as at timet= 1f s.
110 Chapter 4 Analysis of the Matrix Product State Approach to Study Ultrafast Dynamics in
of5%when using a MPS bond dimensionD= 70. The error reduces to<2%when using a MPS bond dimension ofD= 110. We observe the rapid convergence of the MPS results with growing bond dimension that we have already seen for the singly ionized molecules. This convergence is independent of the actual dimension of the complete many-body Hilbert space. The dimension of the complete many-body state for the smallest example hydrogen fluoride molecule is27252(DFCI = 175), where it is for the largest example methane1416368(DFCI = 808). Although these dimensions differ in size, we do not observe the larger Hilbert space requiring significantly larger MPS bond dimensions to find the OBRDM within an error of5%.
The situation changes when discussing inner-valence double ionization (HOMO-3) and double ionization of the core orbital (HOMO-4). Here the MPS approach struggles to represent the many-body states correctly, which we observe in errors of the OBRDM in the range of15%to30%. Increasing the MPS bond dimension reduces the error of the OBRDM in case the molecule was prepared in a state with a double hole in the HOMO-3 orbital. Here we find for an MPS bond dimension ofD = 110an error of the OBRDM in the range of2%to5%. In case the water molecule was prepared in a state with a double hole located in the HOMO-4, a bond dimension ofD= 110is still not sufficient. We need to increase the bond dimension further, which reduces the computational advantage of the MPS approach in comparison to the calculation using the complete many-body state.
The fact that double ionization in core orbitals are difficult to describe using MPS tells us that in these situations the entanglement grows comparably fast. This makes a MPS description of such situations difficult. We need to be careful when describing situations with deeply ionized molecules in the system. This might also be a conse-quence of the used orbital ordering that placed the core orbital at the very left of the MPS representation, leaving it with a large (MPS)distance to energetically large orbitals on the right of the MPS representation. Further studies on the orbital ordering in this situation possibly allow to improve the MPS performance for this situation.
CI Weights
Figure4.12shows the CI weights of the time-dependent many-body state for dica-tionic water H2O2+. The initial state is doubly ionized Hartree–Fock ground state, with the double hole located at the HOMO (Figure4.12a) ), located at the HOMO-1 (Figure 4.12b) ), located at the HOMO-2 (Figure4.12c) ), located at the HOMO-3 (Figure4.12d) ), and located at the MOcore (Figure4.12e) ).
0.0f s 4.0f s 0.0
0.5
1.0 2h0p
3h1p high
t−t0
Pα(t)
HOMO
0.0f s 4.0f s t−t0
HOMO-1
0.0f s 4.0f s t−t0
HOMO-2
0.0f s 4.0f s t−t0
HOMO-3
0.0f s 4.0f s t−t0
MOcore
a) b) c) d) e)
Figure 4.12.: Time-dependent CI weights of the many-body state of dicationic water (H2O2+).
The initial state is a double ionization in the molecular orbital as given in the titles of the plots. The shown CI weights are thetwo-hole-zero-particleweight (2h0p), thethree-hole-one-bodyweight (2h1p) and contributions of higher CI classes (high). In all calculations (based on the complete many-body state and MPS based representation) the state is propagated using the orthogonalized Krylov space method with Krylov space dimensionNKry = 6and time step size
∆t= 1.0as.
The many-body state for outer-valence double ionization has a particular high 2h0p and 3h1p weight (see Figure4.12a) to c) ). In these situations a CI expansion is justified and the system remains in states related to few particle-hole excited configurations.
The settings with a double excitation in the outer-valence orbitals are those we observed good convergence of the MPS representation for (see Figure4.11).
In case of inner-valence double ionization and core orbital double ionization, higher excited configurations become rather important. If the initial state of the water molecule is prepared as doubly ionized HOMO-3, or doubly ionized MOcore Hartree–
Fock state, the weights of the 2h0p and 3h1p configurations reduce. Then the im-portance of the higher excited configurations increases. A simple CI expansion of the many-body state is unable to describe these situations, as multiple particle-hole excitations contribute significantly. This confirms to observations we made for the singly ionized systems already, where ionization of deeper orbitals responds to higher excitations of the molecule. As a consequence of the higher excited initial state, also the dynamics involves higher excited configurations, which we notice in the form of a largePhigh(t)weight.
However, the MPS approach is able to include the weights of the higher CI expan-sion correctly. The CI weight of the MPS representation (bold colors in Figure4.12) correspond in all situations to the CI weights of the complete many-body state (light colors in Figure4.12). Even in situations where the MPS approach looses precision in the OBRDM, the CI classes are accurately found in the MPS representation. This underlines its capabilities to adjusting the representation of the many-body state dynamically.
112 Chapter 4 Analysis of the Matrix Product State Approach to Study Ultrafast Dynamics in
Conclusion
We conclude that the MPS approach enables to represent the many-body state with minor truncation errors in case the molecule has been doubly ionized. However, the convergence needs to be monitored more carefully, as we have observed larger rela-tive errors of the OBRDM if the double hole is located in inner-valence or core orbitals.
Here, the MPS bond dimension needs to be increased compared to the singly ionized situations studied above, which we attribute to the enlarged excitation energy of the initial state, but also the orbital ordering can be responsible for this behavior.
For outer-valence orbitals, we have observed similar convergence performance com-pared to the single ionization situations, where it was easy to reduce the relative error to5%by using bond dimensions ofD = 70. The necessary bond dimension is independent of the dimension of the complete many-body Hilbert space of the problem. This shows that we can gain computational advantages compared to the CI expansions, especially in situations where more electrons and orbitals are involved.
In case of double ionization, the importance of highly excited configurations seems to be increased in comparison to the singly ionized situations.