The implementation of a self-consistent constricted variational density functional theory for the description of excited states

Abstract

We present here the implementation of a self-consistent approach to the calculation of excitation energies within regular Kohn-Sham density functional theory. The method is based on the n-order constricted variational density functional theory (CV(n)-DFT) [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, and F. Wang, J. Chem. Phys. 130, 154102 (2009)]10.1063/1.3114988 and its self-consistent formulation (SCF-CV(∞)-DFT) [J. Cullen, M. Krykunov, and T. Ziegler, Chem. Phys. 391, 11 (2011)]10.1016/j.chemphys.2011.05.021. A full account is given of the way in which SCF-CV(∞)-DFT is implemented. The SCF-CV(∞)-DFT scheme is further applied to transitions from occupied π orbitals to virtual π^* orbitals. The same series of transitions has been studied previously by high-level ab initio methods. We compare here the performance of SCF-CV(∞)-DFT to that of time dependent density functional theory (TD-DFT), CV(n)-DFT and ΔSCF-DFT, with the ab initio results as a benchmark standard. It is finally demonstrated how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(∞)-DFT.

INTRODUCTION

We have recently introduced a variational density functional theory (DFT) scheme for the description of excited states. The new method was based on an n-order constricted variation theory approach (CV(∞)-DFT).¹ Lately, the same method was given a self-consistent formulation and extended to all orders (SCF-CV(∞)-DFT).² The current account provides a full description of the way in which SCF-CV(∞)-DFT is implemented. We apply further SCF-CV(∞)-DFT to a series (T1) of transitions from occupied π orbitals to virtual π^* orbitals. This series has previously been studied³ by an iterative approximate coupled cluster singles, doubles, and triples model, CC3.⁴ We compare here the performance of SCF-CV(∞)-DFT for T1 with other DFT methods,⁵ employing ab initio results³ as the benchmark reference. It is finally discussed how popular DFT based methods applied in the study of excited states such as adiabatic time dependent density functional theory (TD-DFT (Refs. ^{6, 7, 8})) and ΔΔSCF-DFT (Refs. ^{9, 10, 11, 12, 13, 14, 15, 16}) are related through different approximations to SCF-CV(∞)-DFT.

COMPUTATIONAL DETAILS

All calculations were based on DFT as implemented in the Amsterdam Density Functional (ADF) version 2010.¹⁷ Our calculations employed a standard triple-ζ Slater type orbital (STO) basis with one set of polarization functions for all atoms.¹⁸ Use was made of the local density approximation (LDA) in the VWN parametrization¹⁹ as well as the B3LYP and BHLYP exchange hybrid functionals by Becke²⁰ with the correlation functional taken from Lee et al.²¹ All electrons were considered as valence. The accuracy integration parameter (ACINT) for the precision of the numerical integration was set to 5.0. Use was made of a special auxiliary STO basis to fit the electron density in each cycle for an accurate representation of the exchange and Coulomb potentials. The Cartesian coordinates of the 20 (π → π^*) benchmark transitions were taken from the supporting information of Ref. 3. The ground-state geometries of these molecules were optimized at the MP2/6-31G^* level of theory.³

CONSTRICTED VARIATIONAL DFT

Generation of excited state orbitals

In the constricted variational density functional theory, CV(n)-DFT, we construct excited state KS-orbitals by performing a unitary transformation² among occupied²² {ψ_i; i = 1, occ} and virtual {ψ_a; a = 1, vir} ground state orbitals

$$\begin{array}{ccc}\hfill Y\left(\begin{array}{c}{\psi }_{occ}\\ {\psi }_{vir}\end{array}\right)& =& e^U\left(\begin{array}{c}{\psi }_{occ}\\ {\psi }_{vir}\end{array}\right)\hfill \\ & =& \left(\sum _{m=0}^{\infty }\frac{U^m}{m!}\right)\left(\begin{array}{c}{\psi }_{occ}\\ {\psi }_{vir}\end{array}\right)=\left(\begin{array}{c}{\psi }_{occ}^{\prime }\\ {\psi }_{vir}^{\prime }\end{array}\right),\hfill \end{array}$$ (1)

to m = n.

Here ψ_occ and ψ_vir are concatenated column vectors containing the sets of occupied {ψ_i; i = 1, occ} and virtual {ψ_a; a = 1, vir} ground state (reference) KS-orbitals. The resulting ${\psi }_{occ}^{\prime }$ and ${\psi }_{vir}^{\prime }$are concatenated column vectors which contain the sets $\{{\psi }_i^{\prime };i=1,occ\}$ and $\{{\psi }_a^{\prime };a=1,vir\}$ of occupied and virtual excited state orbitals, respectively. The unitary transformation matrix Y is in Eq. 1 expressed in terms of a skew symmetric matrix U as

$$\begin{array}{ccc}\hfill Y& =& e^U=I+U+\frac{U^2}{2}+\cdots \hfill \\ & =& \sum _{m=0}^{\infty }\frac{U^m}{m!}=\sum _{n=0}^{\infty }\frac{{\left(U^2\right)}^m}{2m!}+U\sum _{n=0}^{\infty }\frac{{\left(U^2\right)}^m}{(2m+1)!}.\hfill \end{array}$$ (2)

Here U_ij = U_ab = 0 where “i,j ” refer to the occupied set {ψ_i; i = 1, occ} whereas “a,b” refer to {ψ_a; a = 1, vir}. Further, U_ai are the variational mixing matrix elements that combine virtual and occupied ground state (reference) orbitals in the excited state with U_ai = −U_ia. Thus, the entire matrix U is made up of occ × vir independent elements U_ai. We shall on occasion also consider U as column vector where the index (ai) now runs over pairs of virtual and occupied orbitals. For a given U we can by the help of Eq. 2 generate a set of “occupied” excited state orbitals

$${\psi }_i^{\prime }=\sum _p^{occ+vir}{Y}_{pi}{\psi }_p=\sum _j^{occ}{Y}_{ji}{\psi }_j+\sum _a^{vir}{Y}_{ai}{\psi }_a,$$ (3)

that are orthonormal to order n in U_ai. To second order in U the excited state occupied orbitals are given by¹

$${\psi }_i^{\prime }={\psi }_i+\sum _a^{vir}{U}_{ai}{\psi }_a-\frac{1}{2}\sum _j^{occ}\sum _a^{vir}{U}_{ai}{U}_{aj}{\psi }_j.$$ (4)

On the other hand, to infinite order in U the occupied excited state orbitals take on the form²

$${\psi }_j^{\prime }=\cos \left[\eta {\gamma }_j\right]{\phi }_j^o+\sin \left[\eta {\gamma }_j\right]{\phi }_j^v;j=1,occ.$$ (5)

Here γ_j is defined through the equations

$${\left(V_{vir}^{+}U{V}_{occ}\right)}_{pq}={\gamma }_p{\delta }_{pq};p=1,occ,q=1,occ,$$ (6)

$${\left(V_{vir}^{+}U{V}_{occ}\right)}_{pq}=0;p=occ+1,vir,q=1,occ,$$ (7)

where V_vir and V_occ are two matrices that transforms U into the new matrix T according to $T=V_{vir}^{+}U{V}_{occ}$ with T_pq = 0 for all non-diagonal elements whereas T_pp = γ_pp for the occ × occ diagonal elements. V_occ is²³ the matrix that diagonalizes $D_{jk}^{occ}={\left(U^{+}U\right)}_{jk}$ whereas V_vir diagonalizes $D_{bc}^{vir}={\left(U{U}^{+}\right)}_{bc}$. The pair${\phi }_j^o$, ${\phi }_j^v$ that defines ${\psi }_j^{\prime }$ in Eq. 5 are the set of corresponding orbitals introduced by Amos and Hall²³ and given by

$${\phi }_i^o=\sum _j{\left(V_{occ}\right)}_{ji}{\psi }_{j}i=1,occ,$$ (8)

$${\phi }_i^v=\sum _a^{vir}{\left(V_{vir}\right)}_{ai}{\psi }_{a}i=1,occ,$$ (9)

where ${\sum }_a{\sum }_j{\left(V_{vir}\right)}_{ia}^{+}{U}_{aj}{\left(V_{occ}\right)}_{ji}={\gamma }_i$ for i = 1, occ. Thus in the corresponding orbital representation²³ only one occupied ground state orbital ${\phi }_j^o$ mixes with one corresponding virtual ground state orbital ${\phi }_j^v$ for each occupied excited state orbital ${\phi }_j^{\prime }$. We shall now turn to a discussion of how to determine the variational parameters in U.

Determination of U for CV(2)-DFT

In the simple CV(2)-DFT theory¹ the unitary transformation 1 is carried out to second order in U. We thus obtain the occupied excited state orbitals to second order 4 from which we can generate the excited state Kohn-Sham density ρ^′(1, 1^′) and spin-density s^′(1, 1^′) matrices to second order.^{24, 25}

The expressions^{24, 25} for ρ^′(1, 1^′) and s^′(1, 1^′) make it next possible to write down the corresponding excited state Kohn-Sham energy to second order assuming at the moment real orbitals and U matrix elements. For spin conserving transitions²⁶ the density matrix is given by²⁵

$$\begin{array}{ccc}\hfill {\rho }^{\prime }(1,1^{\prime })& =& {\rho }^{\left(0\right)}(1,1^{\prime })+\Delta {\rho }^{\prime }(1,1^{\prime })\hfill \\ & =& {\rho }^{\left(0\right)}(1,1^{\prime })+\sum _i^{occ}\sum _a^{vir}{U}_{ai}{\psi }_a\left(1^{\prime }\right){\psi }_i\left(1\right)\hfill \\ & & +\sum _i^{occ}\sum _a^{vir}{U}_{ai}{\psi }_a\left(1\right){\psi }_i\left(1^{\prime }\right)\hfill \\ & & +\sum _i^{occ}\sum _a^{vir}\sum _b^{vir}{U}_{ai}{U}_{bi}{\psi }_a\left(1^{\prime }\right){\psi }_b\left(1\right)\hfill \\ & & -\sum _i^{occ}\sum _j^{occ}\sum _a^{vir}{U}_{ai}{U}_{aj}{\psi }_i\left(1^{\prime }\right){\psi }_j\left(1\right),\hfill \end{array}$$ (10)

whereas the excited state energy takes on the form²⁵

$$\begin{array}{ccc}\hfill E_{\mathrm{KS}}\left[{\rho }^{\prime }(1,1^{\prime })\right]& =& E_{\mathrm{KS}}\left[{\rho }^0\right]+\sum _{ai}\sum _{bj}{U}_{ai}{U}_{bj}{A}_{ai,bj}\hfill \\ & & +\sum _{ai}\sum _{bj}{U}_{ai}{U}_{bj}{B}_{ai,bj}+O\left[U^{\left(3\right)}\right],\hfill \end{array}$$ (11a)

where

$$A_{ai,bj}={\delta }_{ab}{\delta }_{ij}({\varepsilon }_a-{\varepsilon }_i)+K_{ai,bj};B_{ai,bj}=K_{ai,jb}.$$ (11b)

Here E_KS[ρ⁰] is the ground state energy and “a,b” run over virtual ground state canonical orbitals whereas “i,j ” run over occupied ground state canonical orbitals, all of the same spin. Further $K_{ai,bj}=K_{ai,bj}^C+K_{ai,bj}^{XC}$ where

$$K_{ai,bj}^C=\int \int {\psi }_a^{*}\left(1\right){\psi }_i\left(1\right)\frac{1}{r_{12}}{\psi }_b\left(2\right){\psi }_j^{*}\left(2\right)d{v}_{1}d{v}_2,$$ (12)

whereas

$$K_{ai,bj}^{XC\left(\mathrm{HF}\right)}=\int \int -{\psi }_a^{*}\left(1\right){\psi }_i^{*}\left(1\right)\frac{1}{r_{12}}{\psi }_b\left(2\right){\psi }_j\left(2\right)d{v}_{1}d{v}_2,$$ (13)

for Hartree Fock exchange correlation and

$$K_{ai,bj}^{XC\left(\mathrm{DFT}\right)}=\int {\psi }_a^{*}\left(1\right){\psi }_i\left(1\right)\left[f^{(+,+)}\left({\rho }^0\right)\right]{\psi }_b\left(1\right){\psi }_j^{*}\left(1\right)d{v}_1,$$ (14)

for spin conserving transitions (i → a; j → b) among orbitals of α spin and

$$K_{ai,bj}^{XC\left(\mathrm{DFT}\right)}=\int {\psi }_a^{*}\left(1\right){\psi }_i\left(1\right)\left[f^{(-,-)}\left({\rho }^0\right)\right]{\psi }_b\left(1\right){\psi }_j^{*}\left(1\right)d{v}_1,$$ (15)

for spin conserving transitions (i → a; j → b) among orbitals of β spin.

Here the kernel f ^{σ, τ} is defined as the second order functional derivative of E_XC

$$f^{\sigma ,\tau }{({\rho }^{+},{\rho }^{-})}_0={\left(\frac{{\delta }^2{E}_{XC}}{\delta {\rho }^{\sigma }\delta {\rho }^{\tau }}\right)}_0\sigma =+,-;\tau =+,-,$$ (16)

where ρ⁺ = 1/2ρ + 1/2s and ρ⁻ = 1/2ρ − 1/2s. Further ρ is the density and s the corresponding spin-density.

For spin-flip excitations (i → a; j → b) where the occupied (i,j) and virtual (a,b) orbitals are of different spin, the energy expression for the excited state is similar²⁷ to that of (11) except that

$$\begin{array}{ccc}\hfill K_{ai,bj}^{XC\left(\mathrm{DFT}\right)}& =& \frac{1}{2}\int {\psi }_a\left(1\right){\psi }_i\left(1\right)[f_0^{+,+}\hfill \\ & & +f_0^{-,-}-2f_0^{+,-}]{\psi }_b\left(1\right){\psi }_j\left(1\right)d{v}_1;K_{ai,bj}^C=0,\hfill \end{array}$$ (17)

where the expressions in Eq. 17 have been derived from non-collinear exchange-correlation theory.^{28, 29, 30}

In CV(2)-DFT (Ref. ¹) we seek points on the energy surface E_KS[ρ^′] such that ΔE_KS[Δρ^′] = E_KS[ρ^′] − E_KS[ρ⁰] represents a transition energy. Obviously, a direct optimization of ΔE_KS[Δρ^′] without constraints will result in ΔE_KS[Δρ^′] = 0 and U = 0. We¹ now introduce the constraint that the electron excitation must represent a change in density Δρ^′ where one electron in Eq. 10 is transferred from the occupied space represented by $\Delta {\rho }_{occ}=-{\sum }_{ija}{U}_{ai}{U}_{aj}^{*}{\psi }_i\left(1^{\prime }\right){\psi }_j^{*}\left(1\right)$ to the virtual space represented by $\Delta {\rho }_{vir}={\sum }_{iab}{U}_{ai}{U}_{bi}^{*}{\psi }_a\left(1^{\prime }\right){\psi }_b^{*}\left(1\right)$. An integration of Δρ_occ and Δρ_vir over all space affords $-\Delta q_{occ}=\Delta q_{vir}={\sum }_{ai}{U}_{ai}{U}_{ai}^{*}$. We shall thus introduce the constraint ${\sum }_{ai}{U}_{ai}{U}_{ai}^{*}=1$. Constructing next the Lagrangian $L=E_{\mathrm{KS}}\left[{\rho }^{\prime }\right]+\lambda (1-{\sum }_{ai}{U}_{ai}{U}_{ai}^{*})$ with λ being a Lagrange multiplier and demanding that L be stationary to any real variation in U results in the eigenvalue equation

$$\left(A^{\mathrm{KS}}+B^{\mathrm{KS}}\right)U^{\left(I\right)}={\lambda }_{\left(I\right)}{U}^{\left(I\right)},$$ (18)

where U^(I) is the eigenvector for excited state I. We can now from Eq. 18 determine the sets of mixing coefficients {U^(I); I = 1, occ × vir} that make L stationary and represent excited states. The corresponding excitation energies are given byλ_(I). This can be seen by multiplying on both sides of Eq. 18 with U^{(I) +} and making use of the constraint and normalization condition U^{(I) +}U^(I) = 1.

Within the Tamm-Dancoff approximation,³¹ Eq. 18 reduces to

$$A^{\mathrm{KS}}{U}^{\left(I\right)}={\lambda }_{\left(I\right)}{U}^{\left(I\right)},$$ (19)

which is identical in form to the equation one obtains from TDDFT in its adiabatic formulation^{6, 7, 8, 9, 10} after applying the same Tamm-Dancoff³¹ approximation.

Excitation energies for SCF − CV(∞) − DFT

From the occupied excited state orbitals of Eq. 15 we can express the electron (ρ^(∞))³² and spin (s^(∞))³² density matrices as well as ρ^+(∞) = 1/2ρ^(∞) + 1/2s^(∞) andρ^−(∞) = 1/2ρ^(∞) − 1/2s^(∞). Starting with a spin-conserving transition, we can without loss of generality assume that it takes place between orbitals of α-spin. Thus, for such a spin conserving transition the excited state orbitals^32a$\{{\psi }_i^{\alpha }\to {\psi }_{i^{\alpha }}^{\prime }=\cos \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right],{\phi }_j^{o_{\alpha }}+\sin \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]{\phi }_j^{v_{\alpha }};j=1,occ/2\}$ are obtained by a unitary transformation to all orders involving the part of the U matrix (U^αα) that according to Eq. 1 mixes occupied ground state orbitals of α-spin with virtual ground state orbitals of α-spin. We can now write^{2, 32a} the change in density within the α manifold due to the excitation as

$$\begin{array}{ccc}\hfill \Delta {\rho }^{\alpha \left(\infty \right)}(1,1^{\prime })& =& \sum _j^{occ/2}{\sin }^2[{\eta }^{\alpha }{\gamma }_j^{\alpha }]\hfill \\ & & \times \left[{\phi }_j^{v_{\alpha }}\left(1^{\prime }\right){\phi }_j^{v_{\alpha }}\left(1\right)-{\phi }_j^{o_{\alpha }}\left(1^{\prime }\right){\phi }_j^{o_{\alpha }}\left(1\right)\right]\hfill \\ & & +\sum _j^{occ/2}\sin \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]\cos \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]\hfill \\ & & \times \left[{\phi }_j^{v_{\alpha }}\left(1\right){\phi }_j^{o_{\alpha }}\left(1^{\prime }\right)+{\phi }_j^{v_{\alpha }}\left(1^{\prime }\right){\phi }_j^{o_{\alpha }}\left(1\right)\right].\hfill \end{array}$$ (20a)

In Eq. 20a the scaling factor η^α is introduced to ensure that Δρ^α(∞)(1, 1^′) represents the transfer of a single electron from the occupied orbital space density $-{\sum }_j^{occ/2}{\sin }^2\left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]{\phi }_j^{o_{\alpha }}\left(1^{\prime }\right){\phi }_j^{o_{\alpha }}\left(1\right)$ to the virtual orbital space density ${\sum }_j^{occ/2}{\sin }^2\left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]{\phi }_j^{v_{\alpha }}\left(1^{\prime }\right){\phi }_j^{v_{\alpha }}\left(1\right)$ or

$$\sum _{j\left(\alpha \right)}^{occ}{\sin }^2\left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]=1.$$ (20b)

Here the constraint of Eq. 20b is a generalization of the corresponding second order constraint ${\sum }_{ai}{U}_{ai}{U}_{ai}^{*}=1$ used to derive Eqs. 18, 19.

The electron (ρ^(∞)) (Refs. ^{32, 33}) and spin (s^(∞)) (Ref. ³²) density matrices as well as ρ^+(∞) = 1/2ρ^(∞) + 1/2s^(∞) = ρ^{+, 0} + Δρ^+(∞) and ρ^−(∞) = 1/2ρ^(∞) − 1/2s^(∞) = ρ^{−, 0} + Δρ^−(∞) allow us next to express the excitation energies^32b as

$$\begin{array}{ccc}\hfill \Delta E^{\left(\infty \right)}& =& E_{\mathrm{KS}}[{\rho }^{+\left(\infty \right)},{\rho }^{-\left(\infty \right)}]-E_{\mathrm{KS}}[{\rho }^{+,0},{\rho }^{-,0}]\hfill \\ & =& \int F_{\mathrm{KS}}^{+}\left[{\rho }^{+,0}+\frac{1}{2}\Delta {\rho }^{+\left(\infty \right)},{\rho }^{-,0}\right.\hfill \\ & & \left.+\frac{1}{2}\Delta {\rho }^{-\left(\infty \right)}\right]\Delta {\rho }^{+\left(\infty \right)}d{v}_1\hfill \\ & & +\int F_{\mathrm{KS}}^{-}\left[{\rho }^{+,0}+\frac{1}{2}\Delta {\rho }^{+\left(\infty \right)},{\rho }^{-,0}\right.\hfill \\ & & \left.+\frac{1}{2}\Delta {\rho }^{-\left(\infty \right)}\right]\Delta {\rho }^{-\left(\infty \right)}d{v}_1\hfill \\ & & +O^{\left[3\right]}(\Delta {\rho }^{+\left(\infty \right)},\Delta {\rho }^{-\left(\infty \right)}),\hfill \end{array}$$ (21)

where for a closed shell ground state molecule with the density ρ⁰, we have ρ^{+, 0} = ρ^{−, 0} = ρ⁰/2. Here Eq. 21 is derived by Taylor expanding³⁴E_KS[ρ^+(∞), ρ^−(∞)] and E_KS[ρ^{+, 0}, ρ^{−, 0}] from the intermediate point (ρ^{+, T}, ρ^{−, T}) = (ρ^{+, 0} + 1/2Δρ^+(∞), ρ^{−, 0} + 1/2Δρ^−(∞)). Further $F_{\mathrm{KS}}^{\sigma }[{\rho }^{+,0}+1/2\Delta {\rho }^{+\left(\infty \right)},{\rho }^{-,0}+1/2\Delta {\rho }^{-\left(\infty \right)})](\sigma =+,-)$ are the Kohn-Sham Fock operators defined with respect to the intermediate point. The expression in Eq. 21 is exact to third order in Δρ^+(∞), Δρ^−(∞), which is usually enough.³⁴ However, its accuracy can be extended to any desired order.³⁴

We get for a spin conserving transition the excitation energy^{32b, 35}

$$\begin{array}{ccc}\hfill \Delta E^{\left(\infty \right)}& =& E_{\mathrm{KS}}^{\infty }[{\rho }^0+\Delta {\rho }^{\left(\infty \right)}]-E_{\mathrm{KS}}\left[{\rho }^0\right]\hfill \\ & =& -\sum _j^{occ/2}{\sin }^2\left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]F_{j^{o_{\alpha }}{j}^{o_{\alpha }}}^{\mathrm{KS},T}\hfill \\ & & +\sum _j^{occ/2}{\sin }^2\left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]F_{j^{v_{\alpha }}{j}^{v_{\alpha }}}^{\mathrm{KS},T}\hfill \\ & & +\sum _j^{occ/2}\cos \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]\sin \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]F_{j^{o_{\alpha }}{j}^{v_{\alpha }}}^{\mathrm{KS},T}\hfill \\ & & +\sum _j^{occ}\cos \left[\eta {\gamma }_j^{\alpha }\right]\sin \left[\eta {\gamma }_j^{\alpha }\right]F_{j^{v_{\alpha }}{j}^{o_{\alpha }}}^{\mathrm{KS},T}\}+O^{\left[3\right]}\left(\Delta {\rho }^{\left(\infty \right)}\right).\hfill \end{array}$$ (22)

Here ${\widehat{F}}^{\mathrm{KS},T}$ is a Kohn-Sham operator defined with respect to the intermediate point (ρ^{+, T}, ρ^{−, T}) and $F_{pq}^{\mathrm{KS},T}$ is a matrix element of this operator involving the two orbitals.

Energy gradient used in optimization of U for SCF − CV(∞) − DFT

We shall now find vectors $U_{\left(\infty \right)}^{\left(I\right)}$ that optimize E_KS[ρ^+(∞), ρ^−(∞)] subject to certain constraints. To that end we will need the energy gradient with respect to variations in U. Considering first a spin conserving transition^37a between orbitals of α-spin we take as a starting point for U^αα that generates $\{{\psi }_i^{\alpha }\to {\psi }_i^{\prime \alpha }=\cos \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]{\phi }_j^{o_{\alpha }}+\sin \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]{\phi }_j^{v_{\alpha }};j=1,occ/2\}$ the elements in $U_{\left(2\right)}^{\left(I\right)}=U^{\left(I\right)}$ that has been found by solving Eq. 19 for a spin conserving transition. To the vector U^(I) corresponds the matrix ${\tilde{U}}^{0,\alpha \alpha }$ and the set $\{{\tilde{\gamma }}_k^{\alpha ,0};k=1,occ\}$. Scaling next ${\tilde{U}}^{0,\alpha \alpha }$ and $\{{\tilde{\gamma }}_k^{\alpha ,0};k=1,occ\}$ by such that ${\sum }_j^{occ}{\sin }^2\left[{\eta }^{\alpha }{\tilde{\gamma }}_j^{\alpha ,0}\right]=1$ affords $U^{0,\alpha \alpha }={\eta }^{\alpha ,}{\tilde{U}}^{0,\alpha \alpha }$ and $\{{\gamma }_k^{\alpha ,0}={\eta }^{\alpha }{\tilde{\gamma }}_k^{\alpha ,0};k=1,occ\}$ where now${\sum }_j^{occ}{\sin }^2\left[{\gamma }_j^{\alpha ,0}\right]=1$. The matrix U^αα is obtained from a CV(2) calculation where U^αα and −U^αα afford the same energy according to Eq. 11a. However in CV(∞) with the energy expression given by Eq. 22, the sign matters through the terms containing $\cos \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]\sin \left[{\eta }^{\alpha }{\gamma }_j^{\alpha }\right]$. As we are dealing with a variational approach, we must pick the sign that affords the lowest energy.

Next, a Taylor expansion of E^(∞)(ρ⁺(U^αα), ρ⁻(U^αα)) from U^{0, αα} to U^α = U^{0, αα} + ΔU^αα affords

$$\begin{array}{ccc}\hfill E^{\left(\infty \right)}\left(U^{\alpha \alpha }\right)& =& E^{\left(\infty \right)}\left(U^{0,\alpha \alpha }\right)+\sum _{ai}{\left(\frac{d{E}^{\left(\infty \right)}}{d{U}_{ai}^{\alpha \alpha }}\right)}_0\Delta U_{ai}^{\alpha \alpha }\hfill \\ & & +\frac{1}{2}\sum _{ai}{\sum _{bj}\left(\frac{d^2{E}^{\left(\infty \right)}}{d{U}_{ai}^{\alpha \alpha }d{U}_{bj}^{\alpha \alpha }}\right)}_0\Delta U_{ai}^{\alpha \alpha }\Delta U_{bj}^{\alpha \alpha }\hfill \\ & =& E^{\left(\infty \right)}\left(U^{0,\alpha \alpha }\right)+\sum _{ai}{\mathbf{g}}_{ai}^{\alpha ,e}\Delta U_{ai}^{\alpha \alpha }\hfill \\ & & +\frac{1}{2}\sum _{ai}\sum _{bj}{\mathbf{H}}_{ai,bj}^{\alpha ,\alpha }\Delta U_{ai}^{\alpha \alpha }\Delta U_{bj}^{\alpha \alpha }+O^{\left(3\right)}\left[\Delta U\right].\hfill \end{array}$$ (23)

A component of the gradient $g_{ai}^{\alpha ,e}$ evaluated at U^{0, αα} reads

$$\begin{array}{ccc}\hfill g_{ai}^{\alpha ,e}\left(U^{0,\alpha \alpha }\right)& =& {\left(\frac{d{E}^{\left(\infty \right)}}{d{U}_{ai}^{\alpha \alpha }}\right)}_0=\sum _{\tau }^{+,-}{\left(\frac{\partial E}{\partial {\rho }^{\tau }}\right)}_0{\left(\frac{\partial {\rho }^{\tau }}{d{U}_{ai}^{\alpha \alpha }}\right)}_0\hfill \\ & =& \sum _{\tau }^{+,-}\int {\widehat{F}}_{\mathrm{KS}}^{\tau }[{\rho }^{+}\left(U^0\right),{\rho }^{-}\left(U^0\right)]{\left(\frac{\partial {\rho }^{\tau }}{d{U}_{ai}^{\alpha \alpha }}\right)}_{0}d{v}_1,\hfill \end{array}$$ (24)

where

$${\rho }^{+}\left(U^0\right)=\frac{1}{2}{\rho }^0+1/2\Delta \rho \left(U^{0,\alpha \alpha }\right)+1/2\Delta s\left(U^{0,\alpha \alpha }\right),$$ (25)

$${\rho }^{-}\left(U^0\right)=\frac{1}{2}{\rho }^0+1/2\Delta \rho \left(U^{0,\alpha \alpha }\right)-1/2\Delta s\left(U^{0,\alpha \alpha }\right).$$ (26)

Here Δρ(U^{0, αα}) is the change in density in going from the ground state with U^{0, αα} = 0 to U^{0, αα} ≠ 0. Further Δs(U^{0, αα}) is the corresponding change in the spin density matrix. The calculation of g and H(U^{0, αα}) in Eq. 23 requires closed form expressions for $d{\rho }^{\tau }\left(U^{\alpha \alpha }\right)/d{U}_{ai}^{\alpha \alpha }$ (Refs. ^37a and ³⁸) and $d^2{\rho }^{\tau }\left(U^{\alpha \alpha }\right)/d{U}_{ai}^{\alpha \alpha }d{U}_{bj}^{\alpha \alpha }$ (τ = +, −).³⁹

Optimization of U for SCF-CV(∞)-DFT

A differentiation of Eq. 23 by ΔU^ααaffords

$${\mathbf{g}}^{e,\alpha }\left(U^{0,\alpha \alpha }\right)+{\mathbf{H}}^{\alpha \alpha }\left(U^{0,\alpha \alpha }\right)\Delta U^{\alpha \alpha }=0,$$ (27)

from which we can find ΔU^σσ iteratively. In the initial steps where |ΔU| ≫ δ_tresh1 the Hessian is calculated approximately by assuming that and H^αα(U^{0, αα}) = ɛ^D with (ɛ^D)_{ai, bj} = δ_ijδ_ab(ɛ_a − ɛ_i). Here ɛ_i, ɛ_a are the orbital energies of the occupied and virtual ground state orbitals, respectively. We thus get for each new step

$$\Delta U^{\alpha \alpha }={\left({\varepsilon }^D\right)}^{-1}{\mathbf{g}}^{e,\sigma }\left(U^{0,\alpha \alpha }\right).$$ (28)

If ${\tilde{U}}^{0,\alpha \alpha }=U^{0,\alpha \alpha }+\Delta U^{0,\alpha \alpha }$ does not satisfy Eq. 20b we introduce a scaling so that ${\widehat{U}}^{0,\alpha \alpha }=\eta {\tilde{U}}^{0,\alpha \alpha }$ satisfy ${\sum }_{j\left(\alpha \right)}^{occ}{\sin }^2\left[{\gamma }_j^{\alpha }\left({\widehat{U}}^{0,\alpha \alpha }\right)\right]=1$. With ${\widehat{U}}^{0,\alpha \alpha }$ satisfying Eq. 20b we finally ensure that ${\widehat{U}}^{0,\alpha \alpha }$ satisfy Tr(U^{0, αα +}U^{K, αα}) = 0 for the excited states K = 1,I–1 that are below the excited state I for which we are optimizing U. This is done by introducing the projection

$$\begin{array}{ccc}\hfill U^{0,\alpha \alpha }& =& {\widehat{U}}^{0,\alpha \alpha }-\sum _{k=1}^{I-1}{U}^{K,\alpha \alpha }Tr\left(U^{K,\alpha \alpha +}{\widehat{U}}^{0,\alpha \alpha }\right)/\hfill \\ & & \times Tr\left(U^{K,\alpha \alpha +}{U}^{K,\alpha \alpha }\right).\hfill \end{array}$$ (29)

After that we go back to Eq. 28 for a new step with U^{0, αα} defined in Eq. 29. When δ_tresh1 > |ΔU| > δ_tresh2 the iterative procedure is resumed by the help of the conjugated gradient technique described by Pople et al.⁴⁰ It is not required in this procedure explicitly to know the Hessian. Instead use is made of the fact that

$$\begin{array}{ccc}\hfill {\mathbf{H}}^{\alpha \alpha }\left(U^{0,\alpha \alpha }\right)\Delta U^{\alpha \alpha }& =& {\mathbf{g}}^{e,\alpha }(U^{0,\alpha \alpha }+\Delta U)\hfill \\ & & -{\mathbf{g}}^{e,\alpha }\left(U^{0,\alpha \alpha }\right)+O^{\left[3\right]}\left(\Delta U\right).\hfill \end{array}$$ (30)

The value for δ_tresh1 is typically 10⁻² whereas δ_tresh2 = 10⁻⁴. Convergence is obtained when the threshold δ_tresh2 is reached. Typically, 20–30 iterations are required to reach δ_tresh1 and 5–10 to reachδ_tresh2. We have also attempted more advanced Hessians for the first part of the optimization such as the one suggested by Fletcher⁴¹ and implemented by Fischer and Almlöf.⁴² However, it was found to be less robust than the simple procedure in Eq. 28. The optimization procedure outlined here for spin-conserving transitions can readily be formulated for spin-flip transitions.

The resolution of singlet and triplet transitions from a closed shell molecule

The outcome of a spin-conserving transition is a state described by a Kohn-Sham determinant with two unpaired electrons of opposite spin. Such a determinant

$${\Psi }_M^{\prime \prime }=\left|{\psi }_1^{\prime \prime }{\psi }_2^{\prime \prime }{\psi }_3^{\prime \prime }...{\psi }_i^{\prime \prime }{\psi }_j^{\prime \prime }...{\psi }_n^{\prime \prime }\right|$$ (31)

has an energy¹³ that is half singlet and half triplet

$$E_M^{\prime }=\frac{1}{2}[E_S^{\prime }+E_T^{\prime }].$$ (32)

On the other hand, the spin-flip transition results in a determinant

$${\Psi }_T^{\prime \prime }=\left|{\psi }_1^{\prime \prime }{\psi }_2^{\prime \prime }{\psi }_3^{\prime \prime }...{\psi }_i^{\prime \prime }{\psi }_j^{\prime \prime }...{\psi }_n^{\prime \prime }\right|,$$ (33)

with the energy of a triplet, $E_T^{\prime \prime }$. We can thus get¹³ the triplet energy from ${\Psi }_T^{\prime \prime }$ as $E_T^{\prime \prime }$ and the singlet energy $E_S^{\prime }$ from ${\Psi }_M^{\prime }$ as¹³

$$E_S^{\prime }\approx 2E_M^{\prime }-E_T^{\prime \prime }=E_S^{\prime }+(E_T^{\prime }-E_T^{\prime \prime }).$$ (34)

Strictly speaking $E_S^{\prime }$ can only be calculated from Eq. 34 if $E_T^{\prime }=E_T^{\prime \prime }$. This will to a good approximation be the case¹³ if all the closed shell orbitals in ${\Psi }_M^{\prime }$ and ${\Psi }_T^{\prime \prime }$ are the same whereas the open shell orbitals have identical spatial parts. To ensure this we optimize U^αα with respect to$E_M^{\prime }$. The resulting orbitals describing ${\Psi }_M^{\prime }$ are subsequently used to construct ${\Psi }_T^{\prime \prime }$ and evaluate $E_T^{\prime \prime }=E_T^{\prime }$ after the required spin adjustment for the open shell orbitals. A slightly more accurate procedure would be to optimize U^αα based on the energy $2E_M^{\prime }-E_T^{\prime \prime }$.

RESULTS AND DISCUSSION

We present here as a first demonstration of our scheme calculations on 20 π → π^* excitation in a series of unsaturated hydrocarbons. The calculations were based on the LDA (Ref. ¹⁹) (VWN), B3LYP (Refs. ²⁰ and ²¹) and BHLYP (Refs. ²⁰ and ²¹) functionals. For each functional results are given for adiabatic TDDFT,^{6, 7, 8, 9, 10} adiabatic TDDFT with the Tamm-Dankoff approximation (TDDFT-TD) and second order constricted variation DFT (CV(2)-TD) within the Tamm-Dankoff approximation according to Eq. 29. The latter two methods should in theory give identical results. We have in addition for each functional results for all order constricted variational DFT (CV(∞) − DFT) (Ref. ²) based on a U matrix obtained with CV(2)-DFT as well as SCF-CV(∞) − DFT in which U is optimized.. We compare further the DFT-based results with the “best estimate” by Schreiber et al.³ obtained from high-level ab initio wave function methods. Also discussed is

$${\lambda }_{\max }={\eta }^{\alpha }{\gamma }_{\max }^{\alpha },$$ (35)

where ${\gamma }_{\max }^{\alpha }$ is the largest eigenvalue in Eq. 16 with respect to U^αα and the scaling factor η^α is introduced to satisfy Eq. 20b. If U^αα is obtained from CV(2)-TD this is indicated by using${\lambda }_{\max }^{\prime }$. On the other hand the use of an optimized U^αα matrix corresponds to${\lambda }_{\max }^{\prime \prime }$.

We deal in the current test set with two groups of excitations. Group A (1,2,5,7,8,9,11,14,17) of Table 1 is associated with π → π^* transitions of the type HOMO − n → LUMO + n (n = 0,1,2) where n = 1,2 applies exclusively to naphthalene. At the simple level of Hückel theory the group A excitations do not have any other π → π^* transition of comparable energy to interact with. Thus, Group A has the potential to be represented by a single orbital transition. Group B (3,4,6,10,12,13,15,16,18,19,20) on the other hand are associated with a pair of π → π^* transitions of the type HOMO → LUMO + n and HOMO − n → LUMO (n = 1,3) that in the case of the simple Hückel theory are degenerate in energy. Thus group B excitations are likely to have contributions from at least two π → π^* transitions.

Table 1.

Vertical singlet excitation¹ energies for a series of π→π^* excitations in unsaturated hydrocarbons based on LDA (VWN).

Molecule	No2 Group	State	Type	Best3	TDDFT4	CV(2)5	CV(2)6	CV(∞)7	λmax8	SCF-CV(∞)9	λmax10
Ethene	1 A11	B1u	π→π*	7.80	7.78	8.48	8.46	8.45	1.181	6.57	1.571
Butadiene	2 A	Bu	π→π*	6.18	5.59	6.16	6.17	6.12	1.169	4.45	1.571
	3 B12	Ag	π→π*	6.55	6.23	6.24	6.23	6.66	0.841	6.56	0.860
Hexatriene	4 B	Ag	π→π*	5.09	5.02	5.03	5.03	5.35	0.787	5.22	1.099
	5 A	Bu	π→π*	5.10	4.50	5.05	5.06	4.94	1.197	3.40	1.571
Octatetraene	6 B	Ag	π→π*	4.47	4.16	4.17	4.17	4.41	0.799	4.30	1.130
	7 A	Bu	π→π*	4.66	3.81	4.34	4.34	4.16	1.212	2.78	1.571
Cyclopropene	8 A	B2	π→π*	7.06	5.99	6.30	6.33	7.55	1.249	5.73	1.571
Cyclopentadiene	9 A	B2	π→π*	5.55	4.97	5.39	5.39	5.88	1.254	4.39	1.571
	10 B	A1	π→π*	6.31	6.03	6.05	6.05	6.45	0.810	6.24	1.040
Norbornadiene	11 A	A2	π→π*	5.34	4.39	4.52	4.52	5.10	1.158	4.32	1.571
	12 B	B2	π→π*	6.11	4.93	4.95	4.95	5.37	0.942	5.01	1.569
Naphthalene	13 B	B3u	π→π*	4.24	4.19	4.20	4.20	4.38	0.788	4.39	0.792
	14 A	B2u	π→π*	4.77	4.05	4.25	4.25	4.71	1.104	3.67	1.571
	15 B	B1g	π→π*	5.99	4.97	4.97	4.97	5.24	0.850	5.23	0.847
	16 B	Ag	π→π*	5.87	5.78	5.80	5.80	5.98	0.854	5.83	1.028
	17 A	B2u	π→π*	6.33	5.86	6.12	6.12	6.09	1.043	5.21	1.570
	18 B	Ag	π→π*	6.67	6.16	6.21	6.20	6.41	0.904	6.35	0.786
	19 B	B3u	π→π*	6.06	5.71	6.22	6.22	6.15	0.730	6.23	0.775
	20 B	B1g	π→π*	6.47	6.14	6.47	6.48	6.37	0.791	6.45	0.829
RMSD A					0.77	0.59	0.59	0.41	…	1.37	…
RMSD B					0.43	0.39	0.39	0.28	…	0.28	…
RMSD13 A+B				–	0.62	0.50	0.50	0.35	…	0.99	…

¹Excitation energies in eV.

²Root mean square deviation.

³Reference ².

⁴TD-DFT without Tamm-Dankoff approximation defined in Eq. 31.

⁵CV(2)-DFT without Tamm-Dankoff approximation, Eq. 28.

⁶CV(2)-DFT with Tamm-Dankoff approximation, Eq. 29 which is equivalent to TDDFT with the Tamm-Dankoff approximation.

⁷Singlet excitation energies for CV(∞) with U taken as solution to Eq. 29.

⁸Based on U taken as solution to Eq. 29 this is the largest eigenvalue from Eq. 16 for this particular excitation.

⁹Excitation energy for CV(∞) with U optimized.

¹⁰Based on optimized an optimized U matrix this is the largest eigenvalue from Eq. 16 for this particular excitation.

¹¹Single orbital transition.

¹²Double orbital transition.

¹³Number used for excitation in text.

LDA results for π → π^* excitations in unsaturated hydrocarbons

Table 1 displays the LDA results. We find that the three second order methods TDDFT, TDDFT-TD, and CV(2)-TD have large root-mean-square deviation (RMSD) values of 0.62 eV, 0.50 eV, and 0.50 eV, respectively. On the average the calculated energies tend to be too small relative to the best estimate³ (BE). We note further that TDDFT-TD and CV(2)-TD as expected give nearly identical results. Any difference of 0.03 eV or smaller stems from the fact that the TDDFT-TD energies are expressed in terms of K integrals according to Eq. 11a whereas excitation energies for CV(2)-TD are derived using Taylor expansions as in Eq. 32. We find that RMSD is 0.59 eV for group A and 0.39 eV for group B in the case of CV(2)/LDA. The corresponding numbers are 0.77 eV and 0.43 eV for TDDFT.

For the CV(∞) scheme, RMSD is reduced to 0.36 eV as the calculated energies on average are slightly higher. In this scheme we add energy terms to all orders in U, with U determined by CV(2) through Eq. 29. Further, RMSD is 0.41 for A and 0.28 for B. Also shown are the maximal scaled eigenvalues ${\lambda }_{\max }^{\prime }$ from Eq. 35 based on U^α with ${\gamma }_{\max }^{\alpha }$ determined from Eq. 16. Here λ_max cannot exceed π/2 as a consequence of Eq. 20b. The maximum value of ${\lambda }_{\max }^{\prime }$ = π/2(1.571) radians corresponds to a single orbital transition.

It follows from Table 1 that the members of group B all have ${\lambda }_{\max }^{\prime }<0.9$. This indicates as expected that these excitations have contributions from more than one π → π^* orbital transition at the CV(2) theory level and thus also for CV(∞). For group A each member only has a significant contribution from one π → π^* orbital transition at the CV(2) level, as expected. Nevertheless, we find ${\lambda }_{\max }^{\prime }\approx 1.2$ rather than ${\lambda }_{\max }^{\prime }$ = π/2(1.571). The reason for this is that other types of transitions than π → π^* contribute, notablyσ → π^*, n → π^* and Rydberg transitions. In terms of groups, RMSD is 0.41 eV for A and 0.28 eV for B in the case of CV(∞).

Turning next to SCF − CV(∞)/LDA in which U^αα is optimized so as to minimize Eq. 32, we note a substantial change in Table 1. Thus, all group A members now have a ${\lambda }_{\max }^{\prime \prime }$value ofπ/2, indicating that they have become single orbital in character. A closer look reveals that the group A members all are represented by pure π → π^* transitions with no significant contribution fromσ → π^*, n → π^* and Rydberg transitions. On the SCF-CV(∞)/LDA level, the group B retains a multi-orbital composition although ${\lambda }_{\max }^{\prime \prime }$ has increased slightly on average. The only two exceptions are transitions 10 and 12 for which ${\lambda }_{\max }^{\prime \prime }$ are 1.3–1.5. Thus, these two transition prefer to be single configurational although a second configuration is available for interaction.

For group A the SCF-CV(∞)/LDA transition energies have dropped as much as 2 eV compared to the CV(∞)/LDA results whereas the energies for the remaining group B transitions hardly have changed. As a result of the substantial change in the predicted group A excitation energies, the RMSD value increases to 0.99 eV for SCF-CV(∞)/LDA. This is exclusively due to group A for which the RMSD is 1.37 eV whereas RMSD is 0.28 eV for B.

To understand the large drop in energy for A we note that the singlet excitation energy for a single orbital pair transition $({\phi }_i^o\to {\phi }_i^v)$ in CV(2)/TD is given by

$$\Delta E_S^{\left(2\right)}={\varepsilon }_{i^v}-{\varepsilon }_{i^o}+2K_{i^o{i}^v,i^o{i}^v}-K_{i^o{\overline{i}}^v,i^o{\overline{i}}^v},$$ (36)

whereas the singlet excitation energy after summing to all orders takes on the form

$$\begin{array}{ccc}\hfill \Delta E_S^{\left(\infty \right)}& =& {\varepsilon }_{i^v}-{\varepsilon }_{i^o}+1/2K_{i^o{i}^o,i^o{i}^o}+1/2K_{i^v{i}^v,i^v{i}^v}\hfill \\ & & -2K_{i^v{i}^v,i^o{i}^o}+K_{{\overline{i}}^v{\overline{i}}^v,i^o{i}^o},\hfill \end{array}$$ (37)

The expression for $\Delta E_S^{\left(2\right)}$ of Eq. 36 appears to be quite different from $\Delta E_S^{\left(\infty \right)}$ of Eq. 37. However, we note that if use is made of pure Hartree-Fock exchange then $K_{i^o{i}^o,i^o{i}^o}^{\mathrm{HF}}=K_{i^v{i}^v,i^v{i}^v}^{\mathrm{HF}}=0$ whereas $K_{i^o{i}^o,i^v{i}^v}^{\mathrm{HF}}=-K_{i^o{i}^v,i^o{i}^v}^{\mathrm{HF}}$ and $K_{i^o{i}^o,{\overline{i}}^v{\overline{i}}^v}^{\mathrm{HF}}=-K_{i^o{\overline{i}}^v,i^o{\overline{i}}^v}^{\mathrm{HF}}$, where $K_{ru,tq}^{\mathrm{HF}}=K_{ru,tq}^C+K_{ru,tq}^{XC\left(\mathrm{HF}\right)}$ with reference to Eqs. 22, 23. Thus, in this case $\Delta E_S^{\left(2\right)\mathrm{HF}}=\Delta E_S^{\left(\infty \right)\mathrm{HF}}$. As a consequence, for Hartree-Fock the excitation energy of a single-orbital transition is fully determined by $\Delta E_S^{\left(2\right)\mathrm{HF}}$ and higher order terms are zero.^{2, 5, 43} However, this is not so for LDA since $K_{i^o{i}^o,i^o{i}^o}^{\mathrm{LDA}}\ne K_{i^v{i}^v,i^v{i}^v}^{\mathrm{LDA}}\ne 0$ and $K_{i^o{i}^o,i^v{i}^v}^{\mathrm{LDA}}\ne -K_{i^o{i}^v,i^o{i}^v}^{\mathrm{LDA}}$ whereas $K_{i^o{i}^o,{\overline{i}}^v{\overline{i}}^v}^{\mathrm{LDA}}\ne -K_{i^o{\overline{i}}^v,i^o{\overline{i}}^v}^{\mathrm{LDA}}$, where $K_{ru,tq}^{\mathrm{LDA}}=K_{ru,tq}^C+K_{ru,tq}^{XC\left(\mathrm{LDA}\right)}$ with reference to Eqs. 22, 24, 25. Thus, for LDA the calculated excitation energies are quite different depending on whether $\Delta E_S^{\left(2\right)}$ or $\Delta E_S^{\left(\infty \right)}$ is used. The same is true for functionals based on the generalized gradient approximation (GGA). In general HF satisfy the symmetry relation $K_{ru,tq}^{\mathrm{HF}}=K_{ru,tq}^C+K_{ru,tq}^{XC\left(\mathrm{HF}\right)}=-K_{rt,uq}^C-K_{rt,uq}^{XC\left(\mathrm{HF}\right)}=-K_{rt,uq}^{\mathrm{HF}}$ since $K_{ru,tq}^{XC\left(\mathrm{HF}\right)}=-K_{rt,uq}^C$ and $K_{rt,uq}^{XC\left(\mathrm{HF}\right)}=-K_{ru,tq}^C$. In LDA and GGA no such relationship exists. Instead we find for π → π^* transitions that $|K_{rt,uq}^{XC\left(\mathrm{LDA}\right)}|\ll |K_{ru,tq}^C|$.

As an example, for the first 1A_g → 1B_u transition of butadiene the terms K_{ππ, ππ} and $K_{{\pi }^{*}{\pi }^{*},{\pi }^{*}{\pi }^{*}}$ that both should be zero have a value of ≈8 eV due to the lack of cancellation of the “self-interaction” Coulomb term by the exchange part. In this case the exchange part is “too small” in absolute terms by one order of magnitude. Further, $K_{\pi \pi ,{\pi }^{*}{\pi }^{*}}^{\mathrm{LDA}}$ and $-K_{\pi {\pi }^{*},\pi {\pi }^{*}}^{\mathrm{LDA}}$ are quite different with $K_{\pi \pi ,{\pi }^{*}{\pi }^{*}}^{\mathrm{LDA}}\approx$ 8 eV and $-K_{\pi {\pi }^{*},\pi {\pi }^{*}}^{\mathrm{LDA}}\approx$ − 2 eV. Likewise $K_{\pi \pi ,{\overline{\pi }}^{*}{\overline{\pi }}^{*}}^{\mathrm{LDA}}$ and $-K_{\pi {\overline{\pi }}^{*},\pi {\overline{\pi }}^{*}}^{\mathrm{LDA}}$ differ since $K_{\pi \pi ,{\overline{\pi }}^{*}{\overline{\pi }}^{*}}^{\mathrm{LDA}}\approx$ 8 eV whereas $-K_{\pi {\overline{\pi }}^{*},\pi {\overline{\pi }}^{*}}^{\mathrm{LDA}}\approx$ − 2 eV. For the last two cases the exchange is “too small” by one order of magnitude for $|K_{rt,uq}^{XC\left(\mathrm{LDA}\right)}|$ to be equal to $|K_{ru,tq}^C|$.

With the Coulomb terms dominating and $K_{\pi \pi ,\pi \pi }^C\approx K_{{\pi }^{*}{\pi }^{*},{\pi }^{*}{\pi }^{*}}^C\approx K_{{\pi }^{*}{\pi }^{*},\pi \pi }^C\approx K_{{\pi }^{*}{\pi }^{*},\overline{\pi }\overline{\pi }}^C$, we get almost cancellation of all K-terms in Eq. 37 with $\Delta E_S^{\left(\infty \right)}$ = 4.5 eV compared to ${\varepsilon }_{{\pi }^{*}}-{\varepsilon }_{\pi }=$ 3.9 eV. For $\Delta E_S^{\left(2\right)}$ of Eq. 36 we have again the Coulomb parts dominating and $K_{\pi \pi ,{\pi }^{*}{\pi }^{*}}^C\approx K_{\pi \pi ,{\overline{\pi }}^{*}{\overline{\pi }}^{*}}^C$. However, with more positive than negative K contributions we get after cancellation that $\Delta E_S^{\left(2\right)}\cong {\varepsilon }_{{\pi }^{*}}-{\varepsilon }_{\pi }+K_{\pi {\pi }^{*},\pi {\pi }^{*}}$ = 6.0 eV. Thus, we have rationalized why a transition described by a single π → π^* excitation in both CV(2)/LDA and SCF-CV(∞)/LDA affords a larger $\Delta E_S^{\left(n\right)}$ for n = 2 than for n = ∞.

The result obtained here where $\Delta E_S^{\left(2\right)}>\Delta E_S^{\left(\infty \right)}$ for π → π^* excitations is typical for transitions $({\phi }_i^o\to {\phi }_i^v)$ where ${\phi }_i^o$ and ${\phi }_i^v$ occupy the same spatial extent such that the densities ${\phi }_i^o{\phi }_i^{o*}$ and ${\phi }_i^v{\phi }_i^{v*}$ have a large overlap. On the other hand for transitions ${\phi }_i^o\to {\phi }_i^v$ where ${\phi }_i^o$ and ${\phi }_i^v$ occupy quite different spatial extents such that the densities ${\phi }_i^o{\phi }_i^{o*}$ and ${\phi }_i^v{\phi }_i^{v*}$ have little or no overlap, we find a reversal to $\Delta E_S^{\left(2\right)}<\Delta E_S^{\left(\infty \right)}$. Examples are charge transfer, σ → π^*, n → π^*, and Rydberg transitions. As already discussed previously,^{5, 10, 43, 44} the reversal comes from the fact that for these types of transitions $1/2K_{i^o{i}^o,i^o{i}^o}+1/2K_{i^v{i}^v,i^v{i}^v}$ in Eq. 37 is far from canceled by $-2K_{i^v{i}^v,i^o{i}^o}+K_{{\overline{i}}^v{\overline{i}}^v,i^o{i}^o}$ such that $\Delta E_S^{\left(\infty \right)}$ becomes large compared to ${\varepsilon }_{i^v}-{\varepsilon }_{i^o}$. On the other hand, in Eq. 36$2K_{i^o{i}^v,i^o{i}^v}-K_{i^o{\overline{i}}^v,i^o{\overline{i}}^v}\approx 0$, so that $\Delta E_S^{\left(2\right)}$ approach ${\varepsilon }_{i^v}-{\varepsilon }_{i^o}$. The reversal is responsible for that TD-DFT and CV(2) affords too low excitation energies for charge transfer, σ → π^*, n → π^*, and Rydberg transitions at the LDA and GGA level of theory. In the present case the reversal explains why σ → π^*, n → π^*, and Rydberg transitions contribute to the group A excitation at the CV(2) level of theory but is absent at the SCF-CV(∞)/LDA level of theory where ${\lambda }_{\max }^{\prime \prime }\approx \pi /2\left(1.571\right)$.

It is worth to mention that for the 10 excitations of group A in Table 1 where ${\lambda }_{\max }^{\prime \prime }$ = 1.57, supplementary ΔSCF/LDA calculations were carried out. The ΔSCF/LDA calculations resulted within 0.05 eV in excitation energies similar to those obtained by RSCF-CV(∞)/LDA and the two methods produced quite similar KS orbitals.

B3LYP and BHLYP results for π → π^* excitations in unsaturated hydrocarbons

Both of the second order methods TDDFT and CV(2)-TD are seen to perform well for B3LYP with RMSD's of 0.41 eV and 0.33 eV, respectively. The two second order schemes fare much worse for BHLYP where the RMSD's are 0.70 eV (TDDFT) and 0.69 eV (CV(2)-TD), respectively. Similar poor results were obtained in the last section for LDA with RMSD's of 0.62 eV (TDDFT) and 0.50 eV (CV(2)-TD), respectively. Thus, RMSD's for the simple second order methods exhibit a strong dependency on the fraction of exact HF-exchange α with a minimum at B3LYP for α = 20%. Most of the error in the second order methods for LDA and BHLYP comes from the group A transitions.

The perturbative all order approach CV(∞) exhibits a weak dependence on the fraction of HF-exchange with RMSD's of 0.35 eV (LDA), 0.27 eV (B3LYP), and 0.31 eV (BHLYP). Thus, although the scheme performs best for B3LYP, CV(∞) is seen to be rather robust with a better performance than the second order theories for all three functionals.

We note for group A that a significant increase in the calculated excitation energies for SCF-CV(∞) as we move to the hybrid functionals and raise the fraction α of HF-exchange in going from LDA (0%) over B3LYP (20%) to BHLYP (50%), Tables 1, 2, 3. The increase in the calculated excitation energies is associated with the fact that only a fraction (1−α) of the excitation energy now is given by the LDA (or GGA) expression $\Delta E_S^{\left(\infty \right)}$ of Eq. 37 whereas the remaining part takes on the form of the HF-equivalent expression $\Delta E_S^{\mathrm{HF}\left(\infty \right)}={\varepsilon }_{i^v}^{\mathrm{HF}}-{\varepsilon }_{i^0}^{\mathrm{HF}}-2K_{i^v{i}^v,i^o{i}^o}^{\mathrm{HF}}+K_{i^v{i}^v,{\overline{i}}^o{\overline{i}}^o}^{\mathrm{HF}}$ since $K_{i^o{i}^o,i^o{i}^o}$ and $K_{i^v{i}^v,i^v{i}^v}$ both are zero for HF. Numerically $\Delta E_S^{\mathrm{HF}\left(\infty \right)}>\Delta E_S^{\left(\infty \right)}$ which accounts for the gradual increase. In terms of RMSD the values for group A decreases gradually from 1.37 eV (LDA) to 0.80 eV (B3LYP) and 0.25 eV (BHLYP). For group B we simply note that RMSD remains relatively constant at the SCF-CV(∞) level of theory with RMSD of 0.28 eV (LDA), 0.21 eV (B3LYP), and 0.27 eV (BHLYP). Thus for the two groups combined we find the RMSD trend 0.99 eV (LDA), 0.59 eV (B3LYP), and 0.26 eV (BHLYP).

Table 2.

Vertical singlet excitation¹ energies for a series of π→π^* excitations in unsaturated hydrocarbons based on B3LYP.

Molecule	No2 Group	State	Type	Best3	TDDFT4	CV(2)5	CV(∞)6	λmax7	SCF-CV(∞)8	λmax9
Ethene	1 A10	B1u	π→π*	7.80	7.73	8.37	8.42	1.209	7.23	1.541
Butadiene	2 A	Bu	π→π*	6.18	5.73	6.26	6.33	1.237	5.08	1.563
	3 B11	Ag	π→π*	6.55	6.76	6.78	6.72	0.879	6.52	0.865
Hexatriene	4 B	Ag	π→π*	5.09	5.67	5.68	5.25	0.791	5.17	0.789
	5 A	Bu	π→π*	5.10	4.69	5.18	5.20	1.273	4.02	1.567
Octatetraene	6 B	Ag	π→π*	4.47	4.82	4.84	4.30	0.800	4.23	0.802
	7 A	Bu	π→π*	4.66	4.02	4.47	4.45	1.301	3.38	1.569
Cyclopropene	8 A	B2	π→π*	7.06	6.34	6.70	7.29	1.201	6.43	1.500
Cyclopentadiene	9 A	B2	π→π*	5.55	5.04	5.43	5.99	1.291	4.86	1.571
	10 B	A1	π→π*	6.31	6.48	6.50	6.34	0.809	6.20	0.879
Norbornadiene	11 A	A2	π→π*	5.34	4.77	4.93	5.51	1.248	4.81	1.534
	12 B	B2	π→π*	6.11	5.49	5.52	5.63	0.956	5.53	0.970
Naphthalene	13 B	B3u	π→π*	4.24	4.41	4.43	4.42	0.784	4.41	0.784
	14 A	B2u	π→π*	4.77	4.35	4.56	4.92	1.130	4.23	1.570
	15 B	B1g	π→π*	5.99	5.54	5.55	5.64	0.903	5.61	0.947
	16 B	Ag	π→π*	5.87	6.13	6.16	5.86	0.831	5.76	0.841
	17 A	B2u	π→π*	6.33	6.12	6.42	6.22	1.049	5.78	1.570
	18 B	Ag	π→π*	6.67	6.82	6.90	6.61	0.861	6.65	0.632
	19 B	B3u	π→π*	6.06	5.91	6.43	6.20	0.747	6.35	0.772
	20 B	B1g	π→π*	6.47	6.32	6.65	6.07	0.859	6.15	0.925
RMSD12 A				–	0.50	0.33	0.32	–	0.80	–
RMSD12 B				–	0.30	0.33	0.20	–	0.21	–
RMSD12 A+B				–	0.41	0.33	0.27	–	0.59	–

¹Excitation energies in eV.

²Number used for excitation in text.

³Reference ².

⁴TD-DFT without Tamm-Dankoff approximation defined in Eq. 31.

⁵CV(2)-DFT with Tamm-Dankoff approximation, Eq. 29 which is equivalent to TDDFT with the Tamm-Dankoff approximation.

⁶Singlet excitation energies for CV(∞) with U taken as solution to Eq. 29.

⁷Based on U taken as solution to Eq. 29 this is the largest eigenvalue from Eq. 16 for this particular excitation.

⁸Excitation energy for CV(∞) with U optimized.

⁹Based on an optimized U matrix this is the largest eigen-value from Eq. 16 for this particular excitation.

¹⁰Single orbital transition.

¹¹Double orbital transition.

¹²Root mean square deviation.

Table 3.

Vertical singlet excitation¹ energies for a series of π→π^* excitations in unsaturated hydrocarbons based on BHLYP.

Molecule	No2 Group	State	Type	Best3	TDDFT4	CV(2)5	CV(∞)6	λmax7	SCF-CV(∞)8	λmax9
Ethene	1 A10	B1u	π→π*	7.80	7.68	8.22	8.45	1.229	8.02	1.571
Butadiene	2 A	Bu	π→π*	6.18	5.91	6.36	6.64	1.292	5.98	1.571
	3 B11	Ag	π→π*	6.55	7.54	7.58	6.96	1.002	6.39	0.869
Hexatriene	4 B	Ag	π→π*	5.09	6.63	6.65	5.08	0.817	4.87	0.802
	5 A	Bu	π→π*	5.10	4.93	5.32	5.54	1.311	4.92	1.571
Octatetraene	6 B	Ag	π→π*	4.47	6.28	5.84	4.01	0.799	3.85	0.805
	7 A	Bu	π→π*	4.66	4.28	4.63	4.77	1.319	4.25	1.570
Cyclopropene	8 A	B2	π→π*	7.06	6.49	6.83	7.15	1.191	7.16	1.570
Cyclopentadiene	9 A	B2	π→π*	5.55	5.15	5.50	6.09	1.311	5.45	1.571
	10 B	A1	π→π*	6.31	7.16	7.19	6.07	0.813	5.85	0.828
Norbornadiene	11 A	A2	π→π*	5.34	5.11	5.31	5.88	1.369	5.35	1.567
	12 B	B2	π→π*	6.11	6.15	6.21	5.93	1.019	5.83	1.021
Naphthalene	13 B	B3u	π→π*	4.24	4.66	4.73	4.26	0.781	4.22	0.774
	14 A	B2u	π→π*	4.77	4.64	4.87	4.81	1.110	4.33	1.111
	15 B	B1g	π→π*	5.99	6.23	6.32	6.48	1.169	6.33	1.318
	16 B	Ag	π→π*	5.87	6.55	6.59	5.46	0.822	5.30	0.820
	17 A	B2u	π→π*	6.33	6.38	6.76	6.29	1.012	6.48	1.120
	18 B	Ag	π→π*	6.67	7.67	7.83	6.85	0.659	6.49	0.810
	19 B	B3u	π→π*	6.06	6.18	6.68	6.23	0.757	6.43	0.771
	20 B	B1g	π→π*	6.47	6.63	6.85	6.24	1.126	6.41	1.255
RMSD12 A				–	0.95	0.94	0.38	–	0.25	–
RMSD12 B				–	0.29	0.23	0.31	–	0.36	–
RMSD12 A+B				–	0.70	0.69	0.35	–	0.31	–

¹Excitation energies in eV.

²Number used for excitation in text.

³Reference ².

⁴TD-DFT without Tamm-Dankoff approximation defined in Eq. 31.

⁵CV(2)-DFT with Tamm-Dankoff approximation, Eq. 29 which is equivalent to TDDFT with the Tamm-Dankoff approximation.

⁶Singlet excitation energies for CV(∞) with U taken as solution to Eq. 29.

⁷Based on U taken as solution to Eq. 29 this is the largest eigenvalue from Eq. 16 for this particular excitation.

⁸Excitation energy for CV(∞) with U optimized.

⁹Based on an optimized U matrix this is the largest eigenvalue from Eq. 16 for this particular excitation.

¹⁰Single orbital transition.

¹¹Double orbital transition.

¹²Root mean square deviation.

CONCLUDING REMARKS

We have here introduced the implementation of the self-consistent constricted variational density functional theory for the description of excited states. The n order constricted variational density functional theory CV(n)-DFT is a generalization of CV(2)-DFT presented in Eq. 29. Combined with the Tamm-Dancoff approximation³¹ CV(2)-DFT affords Eq. 28. Here Eq. 28 is equivalent to the basic equation in TDDFT after the Tamm-Dancoff approximation is applied. In CV(n) we start with the U_I matrix obtained from CV(2)-DFT/TD via Eq. 28. Based on U_I for the Ith excited state we perform in CV(n) a unitary transformation exp [U_I] to order n in U_I on the set of occupied and virtual ground state orbitals under the constraint of Eq. 20b. The result is a new set of occupied and virtual orbitals representing excited state I. This transformation can be carried out to infinite order in n, CV(∞)-DFT. In the self-consistent formulation of CV(∞)-DFT we optimize U_I under the constraints of Eqs. 20b and (41), thus preventing a variational collapse onto the ground state and lower lying excited states. The SCF-CV(∞)-DFT scheme requires on top of a TD-DFT treatment one SCF calculation for each excited state.

We have applied LDA, B3LYP, and BHLYP to a test set (T1) of 20 π → π^* transitions in unsaturated hydrocarbons and compared the calculated excitation energies for TDDFT, CV(2)-TD, CV(∞), and SCF-CV(∞)-DFT to estimates based on high level ab initio calculations.³ The test set T1 was comprised of two groups where all excitations in A could be represented by a single orbital excitation whereas the excitations in B consisted of two or more orbital transitions.

We find that the performance of the second order methods TDDFT and CV(2) is strongly dependent on the fraction α of exact exchange included in the functional. Acceptable results are only obtained with B3LYP where the RMSD's are 0.41 eV (TDDFT) and 0.33 eV (CV(2)-TD), respectively. The poor performance of TDDFT and CV(2) for LDA and BHLYP is primarily to be found among the group A transitions. The CV(∞) scheme where terms to all orders in U are included perturbatively have RMSD's of 0.35 eV (LDA), 0.27 eV (B3LYP), and 0.31 eV (BHLYP). Thus, this scheme is rather insensitive to the fraction α of exact exchange although the best results are obtained for B3LYP. The CV(∞) schemes performs somewhat better for group B than group A. In the case of SCF-CV(∞) we find that the scheme performs well for group B irrespectively of the fraction of HF-exchange with RMSD's of 0.28 eV (LDA), 0.21 eV (B3LYP), and 0.27 eV (BHLYP). On the other hand the performance of SCF-CV(∞) exhibits a strong dependence on α for group A with RMSD's of 1.37 eV (LDA) to 0.80 eV (B3LYP) and 0.25 eV (BHLYP).

Traditionally DFT based studies on excited states have been carried out with either the TD-DFT (Refs. ⁶ and ¹⁰) method or the ΔSCF (Refs. ¹¹ and ¹⁵) scheme. The ΔSCF method³⁶ is as SCF-CV(∞)-DFT variational and can thus describe orbital relaxation due to excitations. However, for a single electron transition it is by design restricted to cases that can be described by a single orbital excitation (i → a). Nevertheless, when applicable, ΔSCF performs well and has been used recently by Van Voorhis et al.⁴⁵ to describe π → π^* transition in conjugated systems using BHLYP or MO6-2× (Ref. ⁴⁶) that contain a high percentage of HF-exchange. The SCF-CV(∞)-DFT method is a natural generalization of the ΔSCF scheme as it can deal with both group A and B excitations. Moreover, it is seen to give good results for BHLYP for both A and B with a RMSD of 0.26 eV. In contrast to the ΔSCF method our scheme ensures that there will be no variational collapse into the ground state. The problem of variational collapse has been discussed in the case of π → π^* transitions for the Hartree method by Davidson and Nitzsche.^36e

Work is now under way to apply SCF-CV(∞)-DFT to charge transfer, σ → π^*, n → π^* and Rydberg transitions as well as transition metal complexes.