π-π

stacking

tackled

with

density

functional

theory

Through

comparison

with

ab

initio

reference

data

we

have

evaluated

the

performance

of

various

density

functionals

for

describing

π-π

interactions

as

a

function

of

the

geometry

between

two

stacked

benzenes

or

benzene

analogs

between

two

stacked

DNA

bases

and

between

two

stacked

Watson

Crick

pairs

Our

main

purpose

is

to

find

a

robust

and

computationally

efficient

density

functional

to

be

used

specifically

and

only

for

describing

π-π

stacking

interactions

in

DNA

and

other

biological

molecules

in

the

framework

of

our

recently

developed

QM/QM

approach

QUILD

In

line

with

previous

studies

most

standard

density

functionals

recover

at

best

only

part

of

the

favorable

stacking

interactions

An

exception

is

the

new

functional

which

correctly

yields

bound

π-stacked

structures

Surprisingly

a

similarly

good

performance

is

achieved

with

the

computationally

very

robust

and

efficient

local

density

approximation

-LRB-

LDA

-RRB-

Furthermore

we

show

that

classical

electrostatic

interactions

determine

the

shape

and

depth

of

the

π-π

stacking

potential

energy

surface

Introduction

Two

major

factors

for

the

structure

and

stability

of

DNA

are

the

hydrogen

bonds

between

two

adjacent

bases

from

opposite

strands

-LRB-

forming

a

Watson-Crick

pair

-RRB-

and

the

π-π

stacking

interactions

between

two

bases

within

each

of

the

two

DNA

strands

The

hydrogen

bonding

interactions

in

DNA

are

well

understood

by

high-level

Density

Functional

Theory

-LRB-

DFT

-RRB-

calculations

-LSB-

-RSB-

They

were

shown

to

be

provided

by

nearly

equally

important

contributions

from

electrostatic

attraction

and

donor-acceptor

orbital

interactions

-LRB-

e.g.

from

an

N

lone

pair

to

an

N-H

σ

orbital

-RRB-

and

to

be

sensitive

to

effects

from

the

molecular

environment

-LRB-

counter-ions

solvent

-RRB-

-LSB-

-RSB-

A

similar

understanding

of

the

π-stacking

interactions

is

currently

lacking

due

to

several

reasons

-LRB-

-RRB-

π-π

stacking

interactions

have

proven

to

be

sensitive

to

the

methodology

used

-LRB-

see

below

-RRB-

-LRB-

-RRB-

π-π

stacking

interactions

have

proven

to

be

very

dependent

on

the

size

of

the

basis

set

used

in

-LRB-

ab

initio

-RRB-

calculations

resulting

in

a

large

basis-set

superposition

error

-LRB-

BSSE

-RRB-

-LRB-

-RRB-

high-level

ab

initio

benchmark

studies

-LRB-

CCSD

-LRB-

T

-RRB-

with

at

least

triple-zeta

basis

sets

-RRB-

have

only

recently

become

available

and

-LRB-

-RRB-

the

notorious

problems

of

DFT

to

correctly

describe

dispersion

interactions

-LSB-

-RSB-

which

constitute

one

of

the

components

involved

in

π-stacking

interactions

-LRB-

see

below

-RRB-

The

latter

problems

of

DFT

resulted

in

potential

energy

surfaces

-LRB-

PES

-RRB-

that

erroneously

lack

any

equilibrium

configuration

i.e.

the

stacked

systems

were

incorrectly

described

as

being

purely

repulsive

Because

of

the

technical

difficulties

involved

in

obtaining

accurate

benchmarks

-LRB-

CCSD

-LRB-

T

-RRB-

using

large

basis

sets

-RRB-

Sponer

Hobza

and

co-workers

have

studied

in

a

ground-breaking

series

of

papers

-LSB-

-RSB-

the

stacking

energies

of

DNA

bases

and

DNA

base

analogs

using

a

variety

of

ab

initio

methods

and

basis

sets

Earlier

studies

showed

that

reasonable

interaction

energies

could

be

obtained

when

employing

the

G

-LRB-

-RRB-

method

which

was

confirmed

by

later

and

more

accurate

studies

-LSB-

-RSB-

in

the

aforementioned

-LRB-

-RRB-

basis

set

the

standard

d-polarization

functions

-LRB-

with

exponent

of

-RRB-

were

replaced

by

more

diffuse

ones

-LRB-

exponent

of

-RRB-

with

the

aim

to

improve

description

of

the

dispersion

attraction

-LSB-

-RSB-

Thus

for

the

anti-parallel

displaced

pyrimidine

dimer

the

G

-LRB-

-RRB-

energy

is

only

kcal

mol

lower

than

the

CCSD

-LRB-

T

-RRB-

G

-LRB-

-RRB-

energy

for

the

anti-parallel

cytosine

dimer

it

is

kcal

mol

higher

than

the

CCSD

-LRB-

T

-RRB-

G

-LRB-

-RRB-

energy

and

for

the

twisted

uracil

dimer

the

G

-LRB-

-RRB-

is

equal

to

the

CCSD

-LRB-

T

-RRB-

G

-LRB-

-RRB-

energy

Although

there

seems

to

be

no

systematic

shift

in

energy

when

comparing

G

-LRB-

-RRB-

with

the

more

elaborate

CCSD

-LRB-

T

-RRB-

stacking

energies

the

stacking

energies

are

always

larger

in

magnitude

than

the

corresponding

CCSD

-LRB-

T

-RRB-

ones

when

using

the

same

basis

set

for

both

methods

Interestingly

in

connection

with

the

quest

for

reliable

ab

initio

benchmarks

preliminary

results

-LSB-

-RSB-

indicate

that

the

recently

developed

spin-component

scaled

-LRB-

-RRB-

yields

systematically

lower

stacking

energies

than

This

suggests

that

is

in

good

agreement

with

the

CCSD

-LRB-

T

-RRB-

results

at

only

a

fraction

of

the

computational

cost

Several

approaches

have

been

proposed

to

remedy

the

dispersion

problem

of

DFT

ranging

from

Time-Dependent

DFT

frameworks

-LSB-

-RSB-

to

more

ad-hoc

procedures

such

as

including

-LRB-

-RRB-

van

der

Waals

terms

in

the

interaction

energy

-LRB-

DFT-D

-RRB-

-LSB-

-RSB-

or

fitting

-LRB-

reparameterizing

-RRB-

the

form

of

the

density

functional

to

some

reference

energies

and/or

distances

A

more

elegant

solution

was

recently

proposed

by

Tozer

and

co-workers

-LSB-

-RSB-

who

analyzed

the

DFT

exchange-correlation

-LRB-

xc

-RRB-

potential

in

comparison

with

the

near-exact

multiplicative

Zhang-Morrison-Parr

-LRB-

ZMP

-RRB-

potential

from

coupled

cluster

Brueckner

doubles

-LRB-

BD

-RRB-

densities

Their

resulting

and

the

related

functional

-LSB-

-RSB-

produces

xc

potentials

that

closely

resemble

those

from

ZMP

As

a

result

and

produce

NMR

chemical

shifts

-LSB-

-RSB-

that

in

general

show

smaller

deviations

from

experimental

values

than

those

from

standard

density

functionals

Furthermore

the

ZMP

potential

from

Brueckner

-LRB-

BD

-LRB-

T

-RRB-

-RRB-

coupled

cluster

densities

gives

dispersion

forces

in

good

agreement

with

near-exact

dispersion

forces

-LSB-

-RSB-

It

is

therefore

to

be

expected

that

the

and

functionals

behave

properly

for

dispersion

forces

as

well

However

the

application

of

these

new

functionals

to

π-stacking

interactions

has

so

far

not

appeared

in

the

literature

Here

we

report

the

results

of

a

systematic

investigation

of

π-stacking

interaction

energies

potential

energy

surfaces

-LRB-

PES

-RRB-

and

stacking

distances

for

dimers

of

among

others

the

five

DNA/RNA

bases

A

long

list

of

standard

as

well

as

recent

density

functionals

-LRB-

including

the

and

functionals

-RRB-

has

been

considered

in

this

study

ranging

from

the

Local

Density

Approximation

-LRB-

LDA

-RRB-

via

the

Generalized

Gradient

Approximation

-LRB-

GGA

-RRB-

and

meta-GGA

to

hybrid

functionals

We

have

also

examined

BHandH

which

was

recently

-LSB-

-RSB-

shown

to

give

reasonable

agreement

with

ab

initio

data

The

DFT

results

are

compared

to

benchmark

data

that

were

taken

from

the

literature

which

in

some

cases

contained

energy

profiles

as

a

function

of

the

torsion

associated

with

the

rotation

of

one

base

with

respect

to

the

other

The

present

investigation

is

part

of

a

larger

undertaking

We

aim

at

developing

an

extended

QM/QM

scheme

designated

QUILD

-LSB-

-RSB-

which

allows

a

-LRB-

complex

-RRB-

molecular

system

to

be

decomposed

into

arbitrary

interpenetrating

domains

such

that

each

type

of

interaction

can

be

described

with

a

different

method

i.e.

the

density

functional

that

performs

best

for

that

type

of

interaction

or

process

for

example

electron-pair

as

well

as

hydrogen

bonding

-LRB-

see

-LSB-

-RSB-

-RRB-

π-π

stacking

-LRB-

LDA

this

work

-RRB-

or

reaction

barriers

-LRB-

OLYP

OPBE

see

-LSB-

-RSB-

-RRB-

This

is

a

pragmatic

-LRB-

of

course

not

a

fundamental

-RRB-

approach

to

cope

with

the

present-day

shortcomings

of

DFT

which

features

density

functionals

that

perform

satisfactorily

for

a

number

but

not

all

types

of

interactions

and

phenomena

The

present

study

focuses

on

identifying

which

DFT

approaches

serve

best

to

describe

the

π-π

stacking

interactions

in

the

above-mentioned

QM/QM

scheme

Furthermore

we

have

analyzed

the

π-stacking

interactions

in

more

detail

by

decomposing

the

interaction

energy

into

three

physically

meaningful

components

namely

electrostatic

energy

Pauli

repulsion

-LRB-

i.e.

the

destabilizing

interactions

between

occupied

orbitals

-RRB-

and

bonding

orbital

interactions

-LSB-

-RSB-

Our

analyses

reveal

among

others

that

the

electrostatic

attraction

between

the

stacked

molecules

is

the

most

important

component

of

the

π-π

stacking

interaction

-LRB-

see

also

-LSB-

-RSB-

-RRB-

which

determines

the

shape

and

depth

of

the

PES

We

have

also

analyzed

the

π-π

interactions

between

two

stacked

Watson

Crick

base

pairs

with

the

purpose

to

differentiate

between

inter

and

intra-strand

stacking

interactions

Methods

Computational

details

All

calculations

were

performed

with

the

Amsterdam

Density

Functional

-LRB-

ADF

-RRB-

program

-LSB-

-RSB-

Molecular

orbitals

-LRB-

MOs

-RRB-

were

expanded

using

a

large

uncontracted

set

of

Slater-type

orbitals

-LSB-

-RSB-

The

basis

is

an

all-electron

basis

of

triple-ζ

quality

augmented

by

two

sets

of

polarization

functions

-LRB-

and

on

H

d

and

f

on

heavy

atoms

-RRB-

An

auxiliary

set

of

s

p

d

f

and

g

STOs

was

used

to

fit

the

molecular

density

and

to

represent

the

Coulomb

and

exchange

potentials

accurately

in

each

SCF

cycle

The

equilibrium

structure

of

individual

benzene

analogs

DNA

bases

and

Watson-Crick

base

pairs

were

optimized

at

the

level

of

DFT

i.e.

using

the

local

density

approximation

-LRB-

LDA

Slater

exchange

and

VWN

-LSB-

-RSB-

correlation

-RRB-

with

non-local

corrections

due

to

Becke

-LSB-

-RSB-

-LRB-

exchange

-RRB-

and

Perdew

-LSB-

-RSB-

-LRB-

correlation

-RRB-

added

self-consistently

which

was

previously

shown

to

be

one

of

the

best

DFT

methods

for

the

accuracy

of

geometries

-LSB-

-RSB-

and

hydrogen-bonding

interactions

-LSB-

-RSB-

The

π-π

stacking

interaction

energies

for

all

density

functionals

were

evaluated

post-SCF

using

LDA

as

SCF

functional

This

procedure

was

recently

shown

to

introduce

an

error

of

only

a

few

tenths

of

a

kcal

mol

-LSB-

-RSB-

Stacking

distances

-LRB-

vertical

separation

-RRB-

and

orientations

-LRB-

twist

angle

-RRB-

were

explored

with

the

various

density

functionals

through

scans

of

the

potential

energy

surface

-LRB-

PES

-RRB-

in

which

the

geometries

of

the

monomers

-LRB-

e.g.

DNA

bases

or

Watson

Crick

base

pairs

-RRB-

were

kept

frozen

PES

scans

as

a

function

of

the

twist

angle

-LRB-

see

below

-RRB-

were

done

using

steps

of

in

case

of

homo-base

stacks

in

the

case

of

hetero-base

stacks

and

for

stacks

of

Watson

Crick

base

pairs

PES

scans

as

a

function

of

the

vertical

separation

-LRB-

see

below

-RRB-

were

done

in

steps

of

Å

The

interaction

energy

-LRB-

see

below

-RRB-

was

studied

as

a

function

of

the

vertical

separation

between

the

two

monomers

in

the

π-stacked

dimer

and

as

a

function

of

the

twist

angle

between

these

monomers

The

latter

is

defined

as

the

right-handed

rotation

of

the

upper

base

around

the

axis

that

runs

through

the

center

of

mass

of

both

the

upper

and

lower

base

and

perpendicular

to

the

plane

of

the

base

-LRB-

see

Scheme

-RRB-

Scheme

of

the

twist

angle

-LRB-

TW

-RRB-

The

black

dot

is

the

center

of

mass

In

the

stacked

DNA

systems

the

twist

angle

of

is

defined

as

that

twist

angle

at

which

the

respective

glycosidic

N-H

bonds

-LRB-

more

precisely

the

N-H

bonds

that

replace

the

glycosidic

N-C

bonds

to

the

sugar

in

the

backbone

-RRB-

are

oriented

in

parallel

Bond-energy

decomposition

The

overall

π-stacking

energy

ΔE

is

made

up

of

two

major

components

-LSB-

-RSB-

-LRB-

Equation

-RRB-

In

this

formula

the

preparation

energy

ΔEprep

is

the

energy

needed

to

deform

the

separate

molecular

fragments

from

their

equilibrium

structure

to

the

geometry

that

they

attain

in

the

overall

molecular

system

The

interaction

energy

ΔEint

is

the

energy

released

when

the

prepared

fragments

are

brought

together

into

the

position

they

have

in

the

overall

molecule

It

is

analyzed

for

our

model

systems

in

the

framework

of

the

KS-MO

model

using

a

Morokuma-type

decomposition

-LSB-

-RSB-

into

electrostatic

interaction

Pauli

repulsion

-LRB-

or

exchange

repulsion

-RRB-

and

-LRB-

attractive

-RRB-

orbital

interactions

-LRB-

Equation

-RRB-

The

term

ΔVelstat

corresponds

to

the

classical

electrostatic

interaction

between

the

unperturbed

charge

distributions

of

the

prepared

-LRB-

i.e.

deformed

-RRB-

fragments

and

is

usually

attractive

The

Pauli-repulsion

ΔEPauli

comprises

the

destabilizing

interactions

between

occupied

orbitals

and

is

responsible

for

the

steric

repulsion

The

orbital

interaction

ΔEoi

in

any

MO

model

and

therefore

also

in

Kohn-Sham

theory

accounts

for

electron-pair

bonding

charge

transfer

-LRB-

i.e.

donor-acceptor

interactions

between

occupied

orbitals

on

one

fragment

with

unoccupied

orbitals

of

the

other

including

the

HOMO-LUMO

interactions

-RRB-

and

polarization

-LRB-

empty-occupied

orbital

mixing

on

one

fragment

due

to

the

presence

of

another

fragment

-RRB-

The

orbital

interaction

energy

can

be

further

decomposed

into

the

contributions

from

each

irreducible

representation

of

the

interacting

system

-LRB-

Equation

-RRB-

using

the

extended

transition

state

-LRB-

ETS

-RRB-

scheme

developed

by

Ziegler

and

Rauk

-LSB-

-RSB-

Results

and

discussion

We

have

analyzed

π-π

stacking

interactions

using

different

systems

that

were

taken

mainly

from

reference

data

present

in

the

literature

We

start

with

the

prototypical

system

for

studying

π-π

stacking

namely

the

benzene

dimer

The

second

set

of

reference

data

was

taken

from

the

supporting

information

of

a

paper

by

Mignon

and

co-workers

-LSB-

-RSB-

who

studied

hydrogen

bonding

and

stacking

in

complexes

of

substituted

benzene

with

pyridine

at

G

-LRB-

-RRB-

with

substituents

X

H

F

Cl

OH

CN

COOH

CHO

and

All

substitutions

of

H

by

another

X

in

benzene

were

done

at

the

para-position

with

respect

to

the

nitrogen

of

pyridine

-LRB-

see

Scheme

-RRB-

The

third

set

of

reference

data

was

taken

from

a

paper

by

Jurecka

and

co-workers

-LSB-

-RSB-

who

investigated

hydrogen

bonding

and

stacking

interactions

in

the

cytosine

dimer

with

and

CCSD

-LRB-

T

-RRB-

methods

using

several

orientations

-LRB-

see

Scheme

-RRB-

to

investigate

the

potential

energy

surfaces

The

fourth

set

of

reference

data

was

taken

from

a

paper

by

Wu

and

Yang

-LSB-

-RSB-

-LRB-

based

on

previous

work

by

Hobza

Sponer

and

co-workers

-LSB-

-RSB-

-RRB-

who

studied

the

dependence

of

the

stacking

energy

of

DNA

bases

on

the

twist

angle

-LRB-

see

Scheme

-RRB-

and

vertical

separation

using

G

-LRB-

-RRB-

calculations

Scheme

of

substituted

benzene

and

pyridineScheme

of

the

stacked

cytosine

dimer

from

Jurecka

and

co-workers

-LRB-

-LSB-

-RSB-

-RRB-

The

upper

base

is

drawn

in

bold

the

center

of

mass

is

indicated

by

a

black

dot

and

the

carbonylic

oxygen

and

the

hydrogen

replacing

the

glycosidic

bond

are

indicated

for

clarityScheme

of

stacked

base

dimers

with

a

twist

angle

of

and

degrees

The

upper

base

is

drawn

in

bold

the

center

of

mass

is

indicated

by

a

black

dot

and

the

hydrogens

replacing

the

glycosidic

bond

are

indicated

for

clarity

Note

that

for

the

homo

dimers

at

zero

twist

angle

the

lower

base

is

hidden

behind

the

upper

base

As

mentioned

already

the

prototypical

system

for

studying

π-π

stacking

interactions

is

posed

by

the

-LRB-

parallel

sandwich

-RRB-

benzene

dimer

Because

of

its

high

symmetry

-LRB-

-RRB-

CCSD

-LRB-

T

-RRB-

energies

using

large

basis

sets

-LRB-

aug-cc-pVQZ

-RRB-

are

available

-LSB-

-RSB-

to

compare

with

-LRB-

see

Table

-RRB-

The

minimum

is

found

at

Å

for

this

system

with

a

stacking

energy

of

kcal

mol

The

standard

density

functionals

are

unable

to

give

a

proper

description

of

this

system

i.e.

they

either

predict

a

purely

repulsive

energy

surface

-LRB-

e.g.

see

Table

-RRB-

or

give

a

shallow

well

-LRB-

e.g.

OLYP

-RRB-

Interestingly

the

functional

predicts

a

minimum

that

is

very

close

-LRB-

Å

-RRB-

to

that

of

the

CCSD

-LRB-

T

-RRB-

method

and

has

a

well

depth

that

is

equally

close

-LRB-

kcal

mol

see

Table

-RRB-

The

related

and

surprisingly

the

LDA

functionals

have

a

minimum

at

the

same

distance

as

CCSD

-LRB-

T

-RRB-

and

but

with

a

somewhat

smaller

well

depth

-LRB-

kcal

mol

-RRB-

Also

the

BHandH

functional

performs

reasonably

well

with

a

similar

equilibrium

distance

but

a

reduced

well

depth

-LRB-

kcal

mol

see

Table

-RRB-

Table

energy

-LRB-

in

kcal

mol

-RRB-

of

the

benzene-benzene

complex

as

a

function

of

the

vertical

separation

Rvert

-LRB-

in

Å

-RRB-

computed

with

various

density

functionalsa

and

CCSD

-LRB-

T

-RRB-

aug-cc-pVQZ

bRvertCCSD

-LRB-

T

-RRB-

0.090.170.924.958.457.2513.147.526.683.5

0.010.040.037.50.030.020.040.090.020.000.03

using

orbitals

and

densities

see

Computational

detailsbFrom

supporting

information

of

-LSB-

-RSB-

Reference

data

from

Mignon

and

co-workers

Scheme

shows

the

structure

of

the

π-stacked

complexes

of

pyridine

with

-LRB-

substituted

-RRB-

benzene

i.e.

with

benzene

fluorobenzene

aminobenzene

chlorobenzene

toluene

phenol

cyanobenzene

benzoic

acid

benzaldehyde

and

nitrobenzene

For

all

of

these

complexes

the

coordinates

were

taken

from

the

supporting

information

of

the

original

paper

by

Mignon

and

co-workers

-LSB-

-RSB-

The

geometry

of

each

dimer

had

been

fully

optimized

and

therefore

the

vertical

separation

is

different

for

each

complex

it

is

found

in

the

range

of

Å

The

interaction

energies

for

several

typical

density

functionals

are

compared

to

the

original

data

in

Table

Consistent

with

previous

reports

the

mean

absolute

deviations

-LRB-

MAD

-RRB-

with

respect

to

the

reference

data

are

much

larger

for

density

functionals

containing

the

-LSB-

-RSB-

exchange

functional

-LRB-

e.g.

MAD

value

kcal

mol

-RRB-

than

for

functionals

based

on

-LSB-

-RSB-

or

PBE

-LSB-

-RSB-

exchange

-LRB-

e.g.

MAD

value

kcal

mol

-RRB-

However

these

standard

density

functionals

all

show

repulsive

interactions

in

contrast

to

the

results

that

indicate

bound

systems

-LRB-

see

Table

-RRB-

Much

better

agreement

with

the

data

is

observed

for

the

and

surprisingly

LDA

functionals

with

MAD

values

of

kcal

mol

-LRB-

see

Table

-RRB-

The

BHandH

functional

recently

-LSB-

-RSB-

shown

to

give

good

agreement

with

-LRB-

T

-RRB-

energies

shows

a

larger

MAD

value

of

kcal

mol

-LRB-

see

Table

-RRB-

Because

of

the

uncertainty

of

ca

kcal

mol

in

the

reference

data

with

respect

to

the

high-level

CCSD

-LRB-

T

-RRB-

data

-LRB-

see

above

-RRB-

it

is

uncertain

which

of

the

four

density

functionals

-LRB-

LDA

or

BHandH

-RRB-

performs

best

However

the

improvement

over

standard

density

functionals

is

obvious

and

promising

Table

energy

-LRB-

in

kcal

mol

-RRB-

of

substituted

benzene-pyridine

complexes

-LRB-

see

Scheme

-RRB-

computed

with

various

density

functionalsa

and

G

-LRB-

-RRB-

bX

F

Cl

OH

CN

COOH

CHO

3.32.05.84.310.84.63.7MADc0.260.250.240.395.339.077.6813.907.896.99aPost-SCF

on

orbitals

and

densities

see

Computational

detailsbFrom

-LSB-

-RSB-

cMean

Absolute

Deviation

-LRB-

in

kcal

mol

-RRB-

of

DFT

energies

from

reference

data

Reference

data

from

Jurecka

and

co-workers

Scheme

shows

the

fourteen

orientations

of

the

stacked

cytosine

dimer

that

have

been

considered

in

the

paper

by

Jurecka

and

co-workers

-LSB-

-RSB-

-LRB-

with

a

vertical

separation

of

Å

for

the

first

orientation

and

Å

for

all

others

-RRB-

The

corresponding

reference

energies

and

our

DFT

interaction

energies

are

collected

in

Table

The

latter

also

contains

the

MAD

values

with

respect

to

two

sets

of

reference

energies

i.e.

the

usual

G

-LRB-

-RRB-

results

-LRB-

-RRB-

as

well

as

the

energies

obtained

after

complete-basis-set

-LRB-

CBS

-RRB-

extrapolation

corrected

for

the

difference

between

and

CCSD

-LRB-

T

-RRB-

with

the

-LRB-

-RRB-

basis

set

-LRB-

-RRB-

Similar

to

what

was

found

for

the

benzene-pyridine

systems

-LRB-

see

above

-RRB-

the

MAD

values

for

standard

density

functionals

are

found

between

and

kcal

mol

resulting

in

many

cases

in

erroneous

repulsive

stacking

interactions

between

the

cytosine

fragments

-LRB-

see

Table

-RRB-

Improved

results

are

again

obtained

with

LDA

and

to

a

lesser

extent

BHandH

which

have

values

-LRB-

i.e.

deviations

with

respect

to

G

-LRB-

-RRB-

-RRB-

of

and

kcal

mol

respectively

and

values

of

and

kcal

mol

respectively

with

the

CCSD

-LRB-

T

-RRB-

data

-LRB-

see

Table

-RRB-

Table

energies

-LRB-

in

kcal

mol

-RRB-

for

several

orientations

of

the

cytosine

dimer

-LRB-

see

Scheme

-RRB-

computed

with

various

density

functionals

and

.22.52.72.32.73.98.011.710.916.510.99.92

2.150.20.30.60.10.61.56.410.39.315.79.38.360.50.60.80.40.81.86.810.89.716.39.88.87

1.3MAD1e0.940.810.601.494.878.357.5813.637.066.03MAD2f0.380.470.720.526.049.528.7514.808.247.21aPost-SCF

on

orbitals

and

densities

see

Computational

detailsbFrom

-LSB-

-RSB-

G

-LRB-

-RRB-

with

complete-basis-set

-LRB-

CBS

-RRB-

extrapolation

i.e.

corrected

for

Δ

-LSB-

CCSD

-LRB-

T

-RRB-

-LRB-

-RRB-

G

-LRB-

-RRB-

-RSB-

eMean

Absolute

Deviation

-LRB-

in

kcal

mol

-RRB-

of

DFT

energies

from

G

-LRB-

-RRB-

datafMean

Absolute

Deviation

-LRB-

in

kcal

mol

-RRB-

of

DFT

energies

from

corrected

for

Δ

-LSB-

CCSD

-LRB-

T

-RRB-

-LRB-

-RRB-

G

-LRB-

-RRB-

-RSB-

Reference

data

from

Wu

and

Yang

In

the

paper

by

Wu

and

Yang

-LSB-

-RSB-

the

potential

energy

surface

-LRB-

PES

-RRB-

was

calculated

as

a

function

of

the

twist

angle

between

two

stacked

bases

-LRB-

see

Scheme

and

methodological

section

for

the

definition

of

the

twist

angle

-RRB-

using

G

-LRB-

-RRB-

which

was

chosen

as

a

reliable

reference

method

based

on

the

paper

by

Hobza

and

co-workers

-LSB-

-RSB-

Although

thymine

was

not

included

by

Wu

and

Yang

we

report

here

the

energy

profiles

for

all

combinations

of

the

five

RNA/DNA

bases

i.e.

thymine

adenine

guanine

cytosine

uracil

-LRB-

see

Scheme

for

the

geometries

-RRB-

The

vertical

separation

is

Å

for

the

C-C

G-C

and

G-G

stacked

base

dimers

and

Å

for

all

other

systems

This

difference

in

vertical

separations

was

chosen

based

on

earlier

papers

by

Sponer

and

co-workers

-LSB-

-RSB-

From

the

tests

with

the

benzene

dimer

-LRB-

Table

-RRB-

the

Mignon

-LRB-

Table

-RRB-

and

Jurecka

reference

data

-LRB-

Table

-RRB-

it

is

already

evident

that

standard

density

functionals

do

not

give

reliable

interaction

energies

which

is

now

shown

to

be

true

also

for

the

energy

profiles

as

function

of

the

twist

angle

Figure

shows

the

PES

for

two

stacked

bases

namely

C-C

and

G-U

which

are

representative

for

all

possible

base

pair

combinations

-LRB-

the

complete

figure

for

all

combinations

of

stacked

bases

can

be

found

in

the

Supporting

Information

-RRB-

Interestingly

although

the

standard

density

functionals

typically

underestimate

the

interaction

energy

the

shape

of

the

PES

is

in

all

cases

highly

similar

to

that

based

on

the

more

accurate

data

This

suggests

that

at

least

trends

and

qualitative

features

in

the

rotational

energy

profile

can

be

correctly

reproduced

with

DFT

which

otherwise

shows

a

-LRB-

functional

and

basis-set

dependent

-RRB-

constant

shift

with

respect

to

the

more

accurate

rotational

profile

Fig.

interaction

-LRB-

in

kcal

mol

-RRB-

as

function

of

the

twist

angle

-LRB-

in

degree

-RRB-

for

π-π

stacks

of

two

bases

C-C

and

G-U

computed

with

various

density

functionals

The

thick

lineswithfilled

circles

triangles

squaresand

diamonds

show

the

results

obtained

with

LDA

BLYP

and

respectively

The

dashed

lines

with

open

squares

trianglesand

diamonds

show

the

results

obtained

with

OLYP

and

respectively

The

bold

linewithcrosses

represents

the

G

-LRB-

-RRB-

reference

data

It

is

however

more

instructive

to

look

at

functionals

that

by

themselves

already

give

accurate

interaction

energies

such

as

LDA

and

BHandH

Figure

shows

the

corresponding

rotational-energy

profiles

again

for

the

stacked

bases

C-C

and

G-U

From

Fig.

it

is

evident

that

the

LDA

and

to

a

lesser

extent

BHandH

are

visually

-LRB-

and

virtually

-RRB-

indistinguishable

from

the

reference

data

Fig.

interaction

-LRB-

in

kcal

mol

-RRB-

as

function

of

the

twist

angle

-LRB-

in

degree

-RRB-

for

π-π

stacks

of

two

bases

C-C

and

G-U

computed

with

various

density

functionals

The

red

blue

and

green

lines

show

the

results

obtained

with

BHandH

and

respectively

The

bold

line

with

crosses

represents

the

G

-LRB-

-RRB-

reference

data

Figure

shows

the

interaction

energy

as

a

function

of

the

vertical

separation

for

both

the

standard

-LRB-

upper

diagrams

-RRB-

and

promising

-LRB-

lower

diagrams

-RRB-

density

functionals

together

with

reference

curve

in

each

of

the

four

diagrams

Not

surprisingly

the

density

functionals

that

gave

larger

errors

for

the

rotational

energy

profile

also

give

larger

errors

for

the

vertical

separation

profiles

almost

all

standard

density

functionals

show

potential

energy

surfaces

that

are

repulsive

or

at

best

weakly

attractive

with

a

very

shallow

minimum

at

elongated

distances

In

contrast

the

promising

functionals

-LRB-

i.e.

LDA

and

BHandH

-RRB-

do

provide

well-defined

minima

with

equilibrium

distances

-LRB-

Å

-RRB-

close

to

the

reference

data

Similar

to

the

uncertainty

in

the

reference

energy

data

of

some

kcal

mol

-LRB-

see

above

-RRB-

we

may

expect

a

similar

uncertainty

for

the

vertical

separation

which

we

very

roughly

estimate

at

Å

The

differences

between

the

equilibrium

distance

of

the

promising

density

functionals

and

that

of

the

reference

falls

well

within

this

estimated

uncertainty

Thus

LDA

and

BHandH

perform

well

not

only

for

rotational

energy

profiles

but

also

for

the

PES

as

a

function

of

the

vertical

separation

Fig.

interaction

-LRB-

in

kcal

mol

-RRB-

as

function

of

the

vertical

separation

-LRB-

in

Å

-RRB-

for

π-π

stacks

of

two

bases

A-A

and

G-G

computed

with

various

density

functionals

The

thick

lines

with

filled

triangles

squares

and

diamonds

in

the

upper

diagrams

show

the

results

obtained

with

BLYP

and

respectively

The

dashed

lines

with

open

squares

triangles

and

diamonds

in

the

upper

diagrams

show

the

results

obtained

with

OLYP

and

respectively

The

red

blue

green

and

orange

lines

in

the

lower

diagrams

that

truly

show

well-defined

minima

are

obtained

with

BHandH

and

LDA

respectively

The

bold

black

line

with

crosses

represents

the

G

-LRB-

-RRB-

reference

data

Decomposition

of

interaction

energy

Now

that

we

have

established

the

reliability

of

the

LDA

and

BHandH

functionals

for

the

interaction

energy

of

stacked

DNA

bases

it

is

interesting

to

study

the

actual

interaction

into

greater

detail

This

is

done

through

a

quantitative

decomposition

of

the

interaction

energy

ΔEint

into

electrostatic

attraction

ΔVelstat

Pauli

repulsion

ΔEPauli

and

orbital

interactions

ΔEoi

-LRB-

see

Methods

-RRB-

Table

provides

the

results

of

the

energy

decomposition

at

for

the

five

π-stacked

homo

dimers

for

each

one

at

a

twist

angle

of

and

-LRB-

which

is

the

lowest-energy

conformation

-RRB-

and

at

two

different

vertical

separations

Note

that

the

lowest-energy

conformation

is

always

achieved

at

a

twist

angle

of

but

that

the

optimal

stacking

distance

-LRB-

vertical

separation

-RRB-

varies

by

Å

between

the

different

π-stacked

homo

dimers

The

energy

changes

by

-LRB-

A-A

-RRB-

-LRB-

C-C

-RRB-

-LRB-

G

G

-RRB-

-LRB-

T-T

-RRB-

and

-LRB-

U-U

-RRB-

kcal

mol

respectively

when

going

from

a

twist

angle

of

to

-LRB-

the

lowest-energy

conformation

-RRB-

This

change

in

energy

is

almost

completely

resulting

from

a

change

in

electrostatics

that

change

by

-LRB-

A-A

-RRB-

-LRB-

C-C

-RRB-

-LRB-

G-G

-RRB-

-LRB-

T-T

-RRB-

and

-LRB-

U-U

-RRB-

kcal

mol

respectively

The

sum

of

Pauli

repulsion

and

orbital

interactions

changes

much

less

by

ca.

kcal

mol

or

less

except

for

T-T

where

the

sum

of

Pauli

repulsion

and

orbital

interactions

changes

by

kcal

mol

due

to

the

influence

of

the

peripheral

methyl

groups

But

also

the

changes

in

each

term

individually

i.e.

either

ΔEPauli

or

ΔEoi

are

much

smaller

-LRB-

typically

to

times

but

in

some

cases

even

more

-RRB-

than

the

changes

in

ΔVelstat

-LRB-

again

only

in

the

case

of

T-T

the

change

in

ΔEPauli

is

nearly

as

large

as

that

in

ΔVelstat

-RRB-

Here

we

note

that

the

same

energy

decomposition

analyses

for

the

five

π-stacked

homo

dimers

at

instead

of

yield

values

that

differ

by

only

kcal

mol

or

less

-LRB-

not

shown

in

Table

-RRB-

Table

of

interaction

energiesa

-LRB-

in

kcal

mol

-RRB-

between

stacked

DNA

bases

in

different

geometries

computed

at

A-AC-CG-GT-TU-UTwist

angle

-LRB-

TW

Rvert

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

ΔVelstat

ΔEoi

ΔEint

Lowest

energy

conformation

-LRB-

TW

Rvert

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

-LRB-

Å

-RRB-

ΔVelstat

ΔEoi

ΔEint

aSee

Computational

details

Additivity

of

interaction

energies

of

stacked

base

pair

dimers

With

the

good

performance

of

for

stacking

interactions

kept

in

mind

we

went

one

step

further

and

looked

also

at

the

interactions

within

stacked

dimersof

hydrogen-bonded

base

pairs

-LRB-

see

Scheme

-RRB-

-LSB-

-RSB-

The

geometry

was

obtained

by

taking

the

geometry

of

one

hydrogen-bonded

base

pair

and

putting

another

one

at

Å

on

top

of

the

former

The

base

pair

dimer

is

stabilized

by

hydrogen

bonding

between

the

bases

within

one

layer

and

stacking

interactions

between

the

bases

in

different

layers

of

the

stack

In

this

case

because

there

are

two

bases

per

layer

there

are

also

interactions

present

between

bases

that

are

in

a

sense

diagonally

on

opposite

sides

and

in

different

layers

-LSB-

-RSB-

the

so-called

cross

terms

For

example

for

the

AT-AT

system

as

shown

in

Scheme

this

refers

to

the

interaction

between

the

thymine

of

the

top

layer

with

the

adenine

of

the

bottom

layer

Scheme

of

stacked

DNA

base

pair

dimer

AT-AT

with

twist

angle

of

The

dot

represents

the

B-DNA

helical

axis

of

rotation

-LSB-

-RSB-

In

order

to

investigate

in

more

detail

the

importance

of

the

various

energy

terms

in

particular

the

cross-terms

we

approximate

the

total

stacking

interaction

between

the

base

pairs

in

the

stacked

base

pair

dimer

by

the

sum

of

four

pairwise

interactions

between

two

bases

i.e.

we

approximate

the

total

-LRB-

stacking

-RRB-

interaction

energy

ΔEint

between

the

layers

AB

and

CD

in

Scheme

by

the

sum

ΔEadd

of

interactions

A-C

B-D

A-D

and

B-C

as

shown

in

Equations

b

Scheme

approximation

for

the

π-π

interaction

between

two

stacked

Watson-Crick

base

pairs

in

terms

of

pairwise

interactions

between

individual

bases

The

remaining

term

ΔEcoop

that

added

to

the

approximate

ΔEadd

yields

again

the

real

stacking

interaction

ΔEint

consists

of

cooperativity

effects

between

the

different

interactions

In

Table

we

report

the

energy

values

resulting

from

the

additivity

scheme

for

the

stacked

AT-AT

GC-GC

and

AU-AU

base

pair

dimers

at

three

different

values

of

the

twist

angle

-LRB-

and

-RRB-

Importantly

the

additive

approximation

ΔEadd

of

the

real

stacking

interaction

ΔEint

largely

accounts

for

the

latter

ΔEcoop

has

in

most

cases

a

positive

sign

and

is

small

compared

to

either

ΔEadd

or

ΔEint

Furthermore

the

dependence

of

the

ΔEcoop

term

on

the

twist

angle

is

much

smaller

than

that

of

the

interaction

energy

ΔEint

For

example

ΔEcoop

varies

from

to

kcal

mol

whereas

ΔEint

varies

from

to

kcal

mol

for

the

stacked

AT-AT

base

pair

dimer

-LRB-

see

Table

-RRB-

Larger

cooperativity

effects

are

found

in

the

case

of

the

stacked

GC-GC

base

pair

dimer

where

ΔEcoop

values

are

found

between

and

kcal

mol

Note

however

that

the

changes

in

ΔEcoop

are

again

small

compared

to

the

changes

in

ΔEint

Table

energy

-LRB-

kcal

mol

-RRB-

between

stacked

DNA

base

pairs

in

terms

of

the

additivity

scheme

computed

at

bSystemTWΔEACΔEBDΔEADΔEBCΔEaddΔEcoopΔEintAT-AT0

aSee

Scheme

bPost-SCF

on

orbitals

densities

The

cross-terms

in

the

additivity

scheme

are

as

important

as

the

non-cross

terms

especially

when

rotating

the

base

pair

dimer

from

a

twist

angle

of

degrees

to

finally

a

twist

angle

of

degrees

At

the

end

point

of

the

A

and

B

base

have

switched

and

the

cross

terms

are

no

longer

AD

and

BC

but

AC

and

BD

this

switch

occurs

after

a

twist

angle

of

For

instance

for

the

AT-AT

base

pair

dimer

at

the

cross

terms

are

stabilizing

-LRB-

kcal

mol

-RRB-

and

dominate

the

ΔEadd

of

kcal

mol

At

the

non-cross

terms

dominate

-LRB-

kcal

mol

-RRB-

with

still

a

substantial

contribution

from

the

cross

terms

-LRB-

kcal

mol

-RRB-

Note

also

the

dominance

of

the

cross

terms

-LRB-

kcal

mol

-RRB-

over

the

non-cross

terms

-LRB-

kcal

mol

-RRB-

for

the

GC-GC

base

pair

dimer

at

This

is

mainly

due

to

favorable

hydrogen-bonding

interactions

between

the

cross

G-G

pair

When

rotating

base

pair

dimers

from

a

twist

angle

of

to

the

cross

terms

in

the

additivity

scheme

reduce

slightly

However

this

is

dominated

by

the

increase

of

favorable

interactions

between

the

stacked

bases

For

instance

for

AT-AT

the

cross

terms

drop

from

to

kcal

mol

but

the

non-cross

terms

increase

from

to

kcal

mol

The

same

trend

is

observed

for

the

AU-AU

and

GC-GC

base

pair

dimers

Again

we

note

that

the

same

energy

decomposition

analyses

for

stacked

DNA

base

pairs

at

instead

of

yield

values

that

differ

by

only

kcal

mol

or

less

-LRB-

not

shown

in

Table

-RRB-

Significance

of

the

observed

trends

for

the

structure

of

DNA

The

stabilizing

interactions

for

the

structure

of

DNA

mainly

originate

from

hydrogen-bonding

interactions

between

bases

in

one

plane

and

the

π-π

stacking

interactions

between

stacked

bases

However

we

have

shown

here

that

in

addition

to

these

terms

there

is

also

an

important

contribution

of

the

cross

terms

that

arise

between

bases

that

are

neither

hydrogen-bonded

within

one

layer

nor

stacked

above

each

other

All

these

interaction

energies

can

now

be

understood

and

quantified

with

density

functional

theory

using

appropriate

functionals

For

the

hydrogen-bonding

interactions

the

and

functionals

seem

to

perform

best

-LSB-

-RSB-

while

for

π-stacking

interactions

and

-LRB-

surprisingly

-RRB-

LDA

work

well

The

experimentally

observed

value

for

the

twist

angle

-LRB-

-RRB-

is

retrieved

in

our

study

on

base

pair

dimers

as

the

orientation

with

optimal

stacking

interactions

We

have

shown

that

this

is

predominantly

determined

by

the

classical

electrostatic

component

of

the

stacking

energy

The

rotation

to

a

twist

angle

of

destroys

part

of

the

favorable

cross

terms

-LRB-

see

above

-RRB-

which

is

however

completely

overcome

by

the

increase

in

favorable

stacking

interactions

between

the

stacked

bases

The

twist

angle

and

vertical

separation

that

are

experimentally

observed

are

of

course

influenced

by

both

the

sugar-phosphate

backbone

and

the

presence

of

solvent

and/or

counter

ions

These

factors

will

be

tackled

in

forthcoming

work

Conclusions

We

have

analyzed

π-π

stacking

interactions

between

two

benzenes

or

benzene

analogs

between

two

DNA

bases

and

between

two

Watson-Crick

base

pairs

using

Density

Functional

Theory

-LRB-

DFT

-RRB-

in

combination

with

large

basis

sets

The

interaction

energies

for

a

large

number

of

density

functionals

have

been

compared

with

ab

initio

reference

data

In

line

with

previous

studies

most

standard

density

functionals

recover

at

best

only

part

of

the

favorable

stacking

interactions

An

exception

is

the

new

functional

which

has

been

constructed

to

closely

resemble

the

near-exact

Zhao

Morrison

Parr

-LRB-

ZMP

-RRB-

exchange-correlation

potential

Thus

not

unexpectedly

gives

a

good

description

for

π-π

stacking

energies

and

potential

energy

surfaces

-LRB-

PES

-RRB-

resulting

in

bound

systems

Surprisingly

however

a

similarly

good

performance

is

achieved

with

the

Local

Density

Approximation

-LRB-

LDA

-RRB-

In

view

of

the

accurate

prediction

of

PESes

and

the

low

computational

cost

we

recommend

the

use

of

either

or

LDA

for

treating

the

π-π

stacking

interactions

in

larger

systems

using

a

QM/QM

approach

-LRB-

e.g.

our

QUILD

approach

which

is

under

development

-LSB-

-RSB-

-RRB-

that

allows

this

type

of

interaction

to

be

separated

from

other

types

of

interaction

such

as

hydrogen

bonding

but

also

regular

covalent

bonds

To

gain

insight

into

the

origin

of

π-π

stacking

interactions

we

have

decomposed

the

interaction

energies

into

the

classical

electrostatic

attraction

Pauli

repulsion

and

orbital

interactions

Interestingly

the

electrostatic

interactions

appear

to

be

the

most

important

factor

that

determines

the

shape

and

depth

of

the

PES

In

the

case

of

two

stacked

Watson-Crick

base

pairs

this

classical

electrostatic

attraction

causes

a

minimum

to

occur

along

the

energy

profile

at

a

twist

angle

of

Furthermore

the

stabilizing

contributions

to

the

stacking

interaction

between

two

Watson-Crick

base

pairs

is

shown

to

originate

from

the

inter-strand

stacking

terms

that

is

from

the

interaction

between

two

bases

that

are

in

different

Watson-Crick

pairs

and

also

not

directly

stacked

on

top

of

each

other

The

above-mentioned

insight

that

electrostatic

attraction

plays

an

essential

role

in

π-π

stacking

interactions

nicely

consolidates

and

extends

the

recent

finding

by

Krapp

Bickelhaupt

and

Frenking

-LSB-

-RSB-

that

classical

electrostatic

attraction

is

essential

for

understanding

trends

in

bond

strength

of

diatomic

molecules

It

is

interesting

to

note

that

qualitative

models

of

both

regular

chemical

bonds

and

those

of

π-π

stacking

interactions

often

completely

ignore

classical

electrostatic

attraction

These

qualitative

models

attempt

to

rationalize

trends

in

bonding

entirely

in

terms

of

orbital

interactions

-LRB-

in

the

case

of

regular

bonds

-RRB-

and

dispersion

interactions

-LRB-

in

the

case

of

π-π

stacking

-RRB-

Electronic

supplementary

material

Below

is

the

link

to

the

electronic

supplementary

material

Supporting

Information

Energy

profiles

for

various

π-π

stacked

systems

as

a

function

of

the

twist

angle

and

the

vertical

separation

computed

with

and

various

density

functionals

-LRB-

PDF

mb

-RRB-

O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

density functional theory
hydrogen bonding

dna stability
dna structure
π-π stacking interactions