Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2 Construction of Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.1 Functionals Based on a Paradigm Two-Electron Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.2 Functionals Based on Reconstruction of 2-RDM in Terms of 1-RDM . . . . . . . . . . . . . 137
2.3 Going Beyond Explicit Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
2.4 Empirical Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3 Predicting Properties of Electronic Systems with Density Matrix Functionals . . . . . . . . . . . 151
3.1 Response Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.2 Ionization Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.3 Fundamental Gap . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 154
4 Optimization of Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5 Time-Dependent RDMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1 Equation of Motion of the 1-RDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2 Time-Dependent Response Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.3 Standard Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4 Alternative Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.5 Phase Including Natural Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
1 Introduction
The most widely used methods in quantum chemistry are either wavefunctionbased or they exploit density functional theory (DFT). The former class of methods
offers high accuracy, but unfavorable scaling of the computational cost with system
size limits their scope of applicability to relatively small systems. Density functional approximations are known to offer a good balance between computational
efficiency and accuracy. Nevertheless, most approximations to DFT are plagued by
generic problems related to the fact that DFT employs a simple local object – the
electron density. Accurate description of the electronic structure of multireference
systems or predicting multiple electronic excitations are examples of problems that
still await satisfactory solutions in DFT. There has recently been growing interest in
approaches embracing simplicity (in the sense that a wavefunction is not involved),
computational efficiency, and versatility of DFT, while lacking the drawbacks.
Functionals of one-electron reduced density matrix (1-RDM) γ, defined for an Nelectron wavefunction Ψ as
γ x; x
0
ð
Þ ¼ N
ð
Á Á Á
ð
Ψ x; x 2 ; . . . ; x N
ð
Þ Ψ
* x
0
; x 2 ; . . . ; x N
ð
Þ dx 2 Á Á Ádx N ;
ð1Þ
where x ¼ (r, s) is a combined spatial and spin coordinate, should, in principle, lead
to formulating methods superior to existing density functional approximations,
especially when static electron correlation effects cannot be neglected. An immediate advantage of using 1-RDM as a main variable instead of the electron density,
ρ, is that the kinetic energy is an explicit functional of γ but not of ρ. Thus, in
126
K. Pernal and K.J.H. Giesbertz
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2 Construction of Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.1 Functionals Based on a Paradigm Two-Electron Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.2 Functionals Based on Reconstruction of 2-RDM in Terms of 1-RDM . . . . . . . . . . . . . 137
2.3 Going Beyond Explicit Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
2.4 Empirical Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3 Predicting Properties of Electronic Systems with Density Matrix Functionals . . . . . . . . . . . 151
3.1 Response Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.2 Ionization Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.3 Fundamental Gap . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 154
4 Optimization of Density Matrix Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5 Time-Dependent RDMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1 Equation of Motion of the 1-RDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2 Time-Dependent Response Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.3 Standard Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4 Alternative Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.5 Phase Including Natural Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
1 Introduction
The most widely used methods in quantum chemistry are either wavefunctionbased or they exploit density functional theory (DFT). The former class of methods
offers high accuracy, but unfavorable scaling of the computational cost with system
size limits their scope of applicability to relatively small systems. Density functional approximations are known to offer a good balance between computational
efficiency and accuracy. Nevertheless, most approximations to DFT are plagued by
generic problems related to the fact that DFT employs a simple local object – the
electron density. Accurate description of the electronic structure of multireference
systems or predicting multiple electronic excitations are examples of problems that
still await satisfactory solutions in DFT. There has recently been growing interest in
approaches embracing simplicity (in the sense that a wavefunction is not involved),
computational efficiency, and versatility of DFT, while lacking the drawbacks.
Functionals of one-electron reduced density matrix (1-RDM) γ, defined for an Nelectron wavefunction Ψ as
γ x; x
0
ð
Þ ¼ N
ð
Á Á Á
ð
Ψ x; x 2 ; . . . ; x N
ð
Þ Ψ
* x
0
; x 2 ; . . . ; x N
ð
Þ dx 2 Á Á Ádx N ;
ð1Þ
where x ¼ (r, s) is a combined spatial and spin coordinate, should, in principle, lead
to formulating methods superior to existing density functional approximations,
especially when static electron correlation effects cannot be neglected. An immediate advantage of using 1-RDM as a main variable instead of the electron density,
ρ, is that the kinetic energy is an explicit functional of γ but not of ρ. Thus, in
126
K. Pernal and K.J.H. Giesbertz