Extending the random phase approximation with renormalized ...

Beyond the Random Phase Approximation: 

Approaching chemical accuracy with renormalized 

adiabatic local density approximations 

Thomas Olsen and Kristian S. Thygesen

Outline 

Correlation energies from the adiabatic connection 

fluctuation-dissipation theorem 

Success and failures of RPA 

Extending RPA with an exchange-correlation kernel 

- The failure of local kernels 

- Introducing non-locality in adiabatic kernels 

Results with a renormalized adiabatic kernel

Correlation energy from ACDFT 

The exact correlation energy in DFT can be written 

E c = −1 

2π∫ 1 

0 

d λ∫ d r d r ' 

e 2 

∣r−r '∣ ∫ ∞ 

d ω[ χ 

0 

λ (r , r ' ;i ω)−χ 0 (r , r ' ;i ω)]



E c = −1 

2π∫ 1 

0 


e 2 

∣r−r '∣ ∫ ∞ 

d ω[ χ 

0 

λ (r , r ' ;i ω)−χ 0 (r , r ' ;i ω)] 

The interacting response function can be obtained from 

TDDFT with a suitable approximation for the xc kernel: 

χ λ (ω)= 

χ 0 (ω) 

λ 0 

1−[λ v+ f xc(ω)]χ 

(ω)



E c = −1 

2π∫ 1 

0 


e 2 

∣r−r '∣ ∫ ∞ 

d ω[ χ 

0 

λ (r , r ' ;i ω)−χ 0 (r , r ' ;i ω)] 

The interacting response function can be obtained from 

TDDFT with a suitable approximation for the xc kernel: 

χ λ (ω)= 

χ 0 (ω) 

λ 0 

1−[λ v+ f xc(ω)]χ 

(ω) 

If f xc is linear in λ, the coupling constant integration can be 

carried out: 

E c=∫ 0 

∞ d ω 

2 π Tr [ v(v+ f x) −1 ln [1−χ 0 (i ω)(v+ f x)]+v χ 0 (i ω)]

RPA Correlation energy 

Neglecting the exchange-correlation kernel gives the 

Random Phase Approximation (RPA) 

E c=∫ 0 

∞ d ω 

2 π Tr [ln [1−χ0 (i ω)v]+v χ 0 (i ω)] 

The expression is implemented in GPAW using a plane 

wave representation for the response function 

Very easy to use... 

from gpaw.xc.rpa_correlation_energy import RPACorrelation 

rpa = RPACorrelation(txt='rpa.txt') 

E_c = rpa.get_rpa_correlation_energy(ecut=300) 

… but significantly more time-consuming than standard KS 

calculations

RPA Correlation energy 

RPA gives an accurate description of van der Waals interactions 

Graphene on metal surfaces 

[T. Olsen and K. S. Thygesen, PRB 87 075111 (2013)]

Atomization energies of small 

molecules 

PBE: MEA – 9 kcal/mol = 0.39 ev 

RPA@PBE: MEA – 10 kcal/mol = 0.43 ev

Cohesive energies of solids 

PBE: MEA – 0.18 ev 

RPA@PBE: MEA – 0.42 ev

Pros and cons in RPA 

The RPA correlation is combined with exact exchange 

and the (first order) self-interaction error vanishes 

Solves the CO puzzle. Correct order of adsorption energies 

on Pt(111) 

Good description of strong static correlation

Pros and cons in RPA 

The RPA correlation is combined with exact exchange 

and the (first order) self-interaction error vanishes 

Solves the CO puzzle. Correct order of adsorption energies 

on Pt(111) 

Good description of strong static correlation 

RPA suffers from large self-correlation errors - the 

correlation energy of a H atom is -0.6 eV 

The atomization energies of small molecules are slightly 

worse than PBE – always underbinds 

The cohesive energies of solids are worse than PBE

Beyond RPA 

It should be possible to improve RPA by including a simple 

exchange-correlation kernel in the response function:

Beyond RPA 

It should be possible to improve RPA by including a simple 

exchange-correlation kernel in the response function: 

The simplest one can think of is the adiabatic LDA kernel 

Furthermore, we only include exchange since 

Such an approximation worsens results significantly! 

[F. Furche and T. van Voorhis, JCP 122 164106 (2005)]

Homogeneous Electron Gas 

To see why ALDA fails one can look at the Fourier transform 

of the correlation hole for the homogeneous electron gas 

[M. Lein, E. K. U. Gross and J. P. Perdew, PRB 61 13431 (2000)] 

r s = 1 r s = 10 

ALDA is not an exact approximation for the HEG! 

The locality implies slow decay at large q = trouble 

[F. Furche and T. van Voorhis, JCP 122 164106 (2005)]


If we make a cutoff at q=2k F equivalent to the truncated kernel 

The correlation energy becomes 

accurate over a wide range of 

densities


If we make a cutoff at q=2k F equivalent to the truncated kernel 

The correlation energy becomes 

accurate over a wide range of 

densities 

The real space correlation hole is 

also much better described than in 

both RPA and pure ALDA 

[T. Olsen and K. S. Thygesen, PRB 86 081103(R) (2012)]

ALDA 

The procedure can be generalized to non-uniform systems by 

Fourier transforming to real space: 

with 

We can then get the kernel for inhomogeneous systems by taking 

[T. Olsen and K. S. Thygesen, PRB 86 081103(R) (2012)]

Spin 

There is not a unique way to generalize to spin-polarized systems 

For a spinpaired system it is straightforward to show that 

=[ ALDA 

f Hx 

f Hxc = 1 

4 ∑σ σ ' f σ σ ' 

Hxc 

The ALDA Hartree-exchange kernel is 

V +2f ALDA 

x [2n↑ ] V 

V V +2f x 

ALDA [2n↓]] 

It is clear that we cannot simply introduce cutoff on the diagonal

Instead we take 

=[ rALDA 

f Hx 

Spin 

V r rALDA 

[n↑+n ↓]+2f x [n↑+n ↓] V r [n ↑+n↓] V r [n↑+n ↓] V r [n ↑+n↓]+2f x 

rALDA [n↑+n ↓]] 

This breaks spin-scaling for the kernel and it cannot be regarded 

as pure exchange 

The choice is not unique!

Instead we take 

=[ rALDA 

f Hx 

Spin 

V r rALDA 

[n↑+n ↓]+2f x [n↑+n ↓] V r [n ↑+n↓] V r [n↑+n ↓] V r [n ↑+n↓]+2f x 

rALDA [n↑+n ↓]] 

This breaks spin-scaling for the kernel and it cannot be regarded 

as pure exchange 

The choice is not unique! 

In contrast to RPA we need to represent the full spin-response 

function – Requires a lot of memory

ALDA 

The rALDA kernel has been implemented in GPAW 

There is no general framework for PAW corrections of two-point 

functions. The implementation uses all-electron density for kernel

ALDA 

The rALDA kernel has been implemented in GPAW 

There is no general framework for PAW corrections of two-point 

functions. The implementation uses all-electron density for kernel 

The method improves absolute correlation energies significantly 

compared to RPA 

[T. Olsen and K. S. Thygesen, PRB 86 081103(R) (2012)] 

Numbers are in kcal/mol = 43 meV

Numbers are in kcal/mol = 43 meV 

rALDA - molecules 

[T. Olsen and K. S. Thygesen, in preparation]

Numbers are in kcal/mol = 43 meV 

rALDA - molecules 

MEA (kcal/mol) 

LDA: 37 

RPA@LDA: 14 

RPA@PBE: 10 

PBE: 9 

SOSEX: 5 

rALDA: 2 


ALDA - solids 

The two-point density makes the rALDA kernel non-periodic 

and one has two sample all unit cells (twice) in bulk systems: 

with 

The two-point density cannot be stored – Implementation 

involves loop over r' and double loop over all unit cells. 

Very slow for solids! 


ALDA - solids 

For semiconductors results are much better than RPA, 

but not for metals... 


ALDA - Static Correlation 

Dissociation of H2 

rALDA results are similar to RPA but offset is much better 


ALDA – van der Waals 

For van der Waals interactions the rALDA kernel gives 

results similar to RPA 

Bilayer graphene 

We have also tested four members of the s22 set of molecular 

dimers where rALDA and RPA produce identical results 


Summary and Outlook 

Compared to RPA, the rALDA kernel significantly improves 

absolute correlation energies 

Atomization energies are significantly improved for small 

molecules and solids 

The rALDA kernel conserves the RPA description of 

dispersive interactions and static correlation 

The method allows for straightforward generalizations to 

renormalized adiabatic GGAs 

- preliminary rAPBE results gives a correlation energy 

for H < 1 meV 

Include correlation part of the adiabatic kernel

Extending the random phase approximation with renormalized ...

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