Batteries

The Batteries application module provides first-principles descriptors for Li-ion cathode and anode materials.

It is available from:

VQkit
 -> Post-processing
 -> [7] Application-oriented analysis
 -> [1] Batteries

The current Batteries menu contains:

[1] Cathode material analysis
    Voltage, capacity, lithiation stability, volume change and specific energy.

[2] Anode material analysis
    Carbon-anode capacity, OCV, formation energy, volume expansion and Li migration barriers.

The battery module is a derived-analysis calculator.

It does not repeat general DOS, band-structure, AIMD, or NEB analysis. Those quantities remain in the common Post-processing categories.

Cathode material analysis

Choose:

[1] Cathode material analysis

The current cathode menu contains:

[1] Average Li insertion voltage
[2] Theoretical specific capacity
[3] Lithiation formation / tie-line energy
[4] Volume change on lithiation / delithiation
[5] Ideal specific-energy estimate
[6] Dilute-limit Li diffusivity estimate
[7] Show equations and scientific reference

[0] Back
[99] Exit VQkit

The cathode workflow is intended for screening DFT-derived Li-ion electrode properties from consistently calculated energies and structures.

[1] Average Li insertion voltage

The intercalation reaction is written as:

\[(x_2-x_1)\mathrm{Li} + \mathrm{Li}_{x_1}\mathrm{Host} \rightarrow \mathrm{Li}_{x_2}\mathrm{Host}\]

The corresponding DFT reaction energy is:

\[\Delta E = E(x_2) - \left[ E(x_1) + (x_2-x_1)E(\mathrm{Li}) \right]\]

The average insertion voltage is:

\[V_{\mathrm{avg}} = -\frac{\Delta E} {(x_2-x_1)z}\]

where:

  • \(x_1\) is the initial Li content;

  • \(x_2\) is the final Li content;

  • \(E(x_1)\) is the energy of \(\mathrm{Li}_{x_1}\mathrm{Host}\);

  • \(E(x_2)\) is the energy of \(\mathrm{Li}_{x_2}\mathrm{Host}\);

  • \(E(\mathrm{Li})\) is the Li-metal energy per atom;

  • \(z\) is the ionic charge of Li.

For Li:

\[z = 1\]

VQkit asks:

Initial Li content x1 [0]:
Final Li content x2 [1]:
E(Li_x1 Host) in eV/f.u.:
E(Li_x2 Host) in eV/f.u.:
E(Li metal) in eV/atom:
Li ionic charge z [1]:

The output includes:

Reaction energy
Average voltage
Reference: Li/Li+

Because the input energies are supplied in eV, the electron-charge conversion for monovalent Li reduces naturally to volts.

Voltage interpretation

A positive average voltage means the insertion reaction is thermodynamically favorable relative to the Li-metal reference under the adopted 0 K approximation.

The calculated quantity is an average voltage between two compositions.

It is not automatically the full experimental voltage profile.

[2] Theoretical specific capacity

VQkit evaluates:

\[Q_{\mathrm{th}} = \frac{nF} {3.6M}\]

where:

  • \(n\) is the number of transferred electrons per formula unit;

  • \(F\) is the Faraday constant;

  • \(M\) is the molar mass in g/mol.

VQkit uses:

\[F = 96485.33212\ \mathrm{C\,mol^{-1}}\]

The resulting capacity is reported in:

mAh/g

Mass input

VQkit provides:

[1] Enter cathode chemical formula
[2] Enter molar mass directly

For formula input, examples include:

LiFePO4
LiNi0.5Mn0.5O2
Li2MnSiO4

The built-in parser supports ordinary element/count notation including decimal stoichiometry.

Examples:

LiFePO4
Li2MnSiO4
LiNi0.5Mn0.5O2

The current parser intentionally rejects ambiguous or unsupported forms such as:

  • variable-x formulas;

  • fractional notation such as 1/3;

  • hydration dots;

  • parentheses;

  • syntax that cannot be parsed unambiguously.

For unsupported compositions, enter the molar mass directly.

Capacity example

For one transferred electron per formula unit:

\[Q_{\mathrm{th}} = \frac{F}{3.6M}\]

A lower molar mass gives a higher theoretical gravimetric capacity for the same electron count.

[3] Lithiation formation / tie-line energy

For an intermediate composition \(x\) between endpoint compositions \(x_1\) and \(x_2\), VQkit evaluates the energy relative to the straight tie line.

Define:

\[f = \frac{x-x_1} {x_2-x_1}\]

The tie-line energy is:

\[E_{\mathrm{tie}} = (1-f)E(x_1) + fE(x_2)\]

The formation/tie-line energy is:

\[\Delta E_{\mathrm{form}} = E(x) - E_{\mathrm{tie}}\]

VQkit asks:

Lower endpoint composition x1 [0]:
Upper endpoint composition x2 [1]:
Intermediate composition x:
E(x1) in eV/f.u.:
E(x2) in eV/f.u.:
E(x) in eV/f.u.:

Interpretation

If:

\[\Delta E_{\mathrm{form}} \le 0\]

VQkit reports that the state lies on or below the endpoint tie line.

If:

\[\Delta E_{\mathrm{form}} > 0\]

the state lies above the endpoint tie line.

This is useful for analyzing relative lithiation stability between two endpoint compositions.

It is not a full convex-hull phase-stability calculation across all competing phases.

[4] Volume change on lithiation / delithiation

VQkit evaluates:

\[\Delta V(\%) = 100 \frac{ V_{\mathrm{final}} - V_{\mathrm{initial}} }{ V_{\mathrm{initial}} }\]

VQkit asks:

Initial cell volume (A^3):
Final cell volume (A^3):

The sign convention is:

positive = expansion
negative = contraction

Volume change is important because large structural expansion can contribute to:

  • mechanical degradation;

  • particle cracking;

  • interfacial instability;

  • loss of electrical contact.

The percentage should be calculated using consistently defined cells.

[5] Ideal specific-energy estimate

VQkit estimates ideal active-material specific energy as:

\[E_{\mathrm{specific}} = V_{\mathrm{avg}} Q_{\mathrm{specific}}\]

with:

  • voltage in V;

  • capacity in mAh/g.

The numerical result is reported in:

Wh/kg

because:

\[1\ \mathrm{V} \times 1\ \mathrm{mAh/g} = 1\ \mathrm{Wh/kg}\]

VQkit asks:

Average voltage (V):
Specific capacity (mAh/g):

The result is explicitly an:

ideal active-material value

It is not a full-cell energy density.

A practical cell would also include:

  • counter electrode;

  • electrolyte;

  • current collectors;

  • separator;

  • binder;

  • conductive additives;

  • packaging;

  • inactive excess materials.

[6] Dilute-limit Li diffusivity estimate

VQkit implements the hopping expression:

\[D = a^2 g f x \nu \exp \left( -\frac{E_a}{k_BT} \right)\]

where:

  • \(a\) is Li hop distance;

  • \(g\) is a geometry factor;

  • \(f\) is a correlation factor;

  • \(x\) is the defect concentration/fraction;

  • \(\nu\) is attempt/effective vibrational frequency;

  • \(E_a\) is the migration barrier;

  • \(k_B\) is Boltzmann’s constant;

  • \(T\) is temperature.

VQkit uses:

\[k_B = 8.617333262145\times10^{-5} \ \mathrm{eV/K}\]

Input

VQkit asks:

Li hop distance a (angstrom):
Geometry factor g [1]:
Correlation factor f [1]:
Diffusion-mediating defect fraction x:
Attempt/effective vibrational frequency nu (s^-1) [1e13]:
Activation barrier Ea (eV):
Temperature (K) [300]:

The hop distance is converted from Angstrom to meters before evaluating the expression.

The final diffusivity is reported in:

m^2/s

Validity

This expression is intended for the dilute-defect hopping limit.

VQkit explicitly warns that it is not appropriate without further treatment for:

  • strongly interacting defects;

  • concentrated Li/vacancy populations;

  • strongly correlated hopping;

  • two-phase transport;

  • collective diffusion mechanisms.

The input migration barrier should normally come from a physically validated NEB or equivalent transition-state calculation.

[7] Cathode equations and reference

The equations menu displays the core relations for:

  • intercalation reaction energy;

  • average insertion voltage;

  • theoretical capacity;

  • tie-line formation energy;

  • volume change;

  • dilute-limit Li diffusivity.

The current code cites:

Meng and Arroyo-de Dompablo, Energy & Environmental Science, 2009, 2, 589-609, DOI: 10.1039/b901825e.

Anode material analysis

Choose:

[2] Anode material analysis

The current anode module is specifically oriented toward Li storage in carbon-based anodes.

The menu contains:

[1] Theoretical specific capacity
[2] Average Li intercalation voltage / OCV
[3] Carbon-anode formation energy
[4] Volume expansion on lithiation
[5] Li diffusion-barrier summary
[6] Show equations and scientific notes

[0] Back
[99] Exit VQkit

[1] Anode theoretical specific capacity

The anode-specific capacity uses the same general expression:

\[Q_{\mathrm{th}} = \frac{n_{\mathrm{Li}}F} {3.6M_{\mathrm{host}}}\]

For the anode workflow, VQkit instructs the user to enter the:

DELITHIATED host formula

and explicitly exclude inserted Li from the host molar mass.

Examples displayed by VQkit include:

C6 for graphite host
C2 for SWCNT host

For graphite:

LiC6 -> approximately 372 mAh/g

For a hypothetical LiC2 basis:

LiC2 -> approximately 1116 mAh/g

These values follow directly from the host mass and inserted Li count.

VQkit asks:

Host formula or molar mass
Maximum inserted Li per host formula unit

The result is reported using the mass of the delithiated host.

[2] Average Li intercalation voltage / OCV

For the carbon-anode workflow, VQkit evaluates the same Li insertion reaction energy:

\[\Delta E = E(x_2) - E(x_1) - (x_2-x_1)E(\mathrm{Li})\]

and:

\[V_{\mathrm{avg}} = -\frac{\Delta E} {x_2-x_1}\]

because Li is monovalent:

\[z = 1\]

VQkit asks:

Initial Li content x1 [0]:
Final Li content x2 [1]:
E(Li_x1 Host) in eV for the chosen formula/cell:
E(Li_x2 Host) in eV for the same formula/cell:
E(bcc Li metal) in eV/atom:

The result is reported as:

Average OCV ... V vs Li/Li+

The implemented approximation is:

\[\Delta G \approx \Delta E\]

at 0 K.

This means entropic, vibrational, configurational, and finite-temperature contributions are not included automatically.

[3] Carbon-anode formation energy

For carbon hosts, VQkit implements:

\[\Delta E_f = \frac{6}{n_C} \left[ E(G+n\mathrm{Li}) - E(G) - nE(\mathrm{Li}) \right]\]

where:

  • \(E(G+n\mathrm{Li})\) is the lithiated carbon-host energy;

  • \(E(G)\) is the pristine carbon-host energy;

  • \(E(\mathrm{Li})\) is Li-metal energy per atom;

  • \(n\) is the number of inserted Li atoms;

  • \(n_C\) is the number of carbon atoms in the host cell.

The factor:

\[\frac{6}{n_C}\]

normalizes the formation energy to a C6 host basis.

VQkit reports both:

Raw insertion term
Delta E_f (C6)

Interpretation

If:

\[\Delta E_f < 0\]

VQkit reports favorable Li insertion relative to the selected pristine-carbon and Li-metal references.

If:

\[\Delta E_f > 0\]

the insertion is unfavorable relative to those same references.

Input

VQkit asks:

E(G+nLi) lithiated carbon host (eV/cell):
E(G) pristine carbon host (eV/cell):
E(Li metal) (eV/atom):
Number of inserted Li atoms n:
Number of carbon atoms n_C in host cell:

[4] Volume expansion on lithiation

The anode workflow uses:

\[\Delta V = \frac{ V(G+n\mathrm{Li}) - V(G) }{ V(G) }\]

VQkit reports both:

  • fractional volume change;

  • percentage volume change.

The percentage form is:

\[100\Delta V\]

The sign convention is:

positive = expansion
negative = contraction

Input:

Pristine host volume V(G) (A^3):
Lithiated host volume V(G+nLi) (A^3):

Large volume expansion can be a disadvantage for repeated charge/discharge cycling.

[5] Li diffusion-barrier summary

This option summarizes one or more Li migration barriers already obtained from DFT/NEB calculations.

VQkit asks for comma-separated barriers:

Li diffusion barriers in eV
(comma separated, e.g. 0.12,0.72,0.92):

The module reports:

Number of paths
Minimum barrier
Mean barrier
Maximum barrier

Mathematically:

\[E_{\min} = \min(E_{a,i})\]
\[E_{\max} = \max(E_{a,i})\]
\[\bar{E}_a = \frac{1}{N} \sum_i E_{a,i}\]

Negative migration barriers are rejected as invalid input.

Interpretation

Lower migration barriers generally indicate easier Li motion along the evaluated pathway.

However, this option is only a barrier summary.

It does not automatically calculate:

  • diffusion coefficients;

  • hopping prefactors;

  • path degeneracy;

  • correlation factors;

  • defect concentrations;

  • macroscopic ionic conductivity.

For a kinetic diffusivity estimate, use the cathode dilute-limit diffusivity calculator where its assumptions are appropriate.

[6] Anode equations and scientific notes

The current code displays equations for:

  • theoretical specific capacity;

  • average Li intercalation voltage;

  • carbon-anode formation energy;

  • volume expansion;

  • Li migration-barrier comparison.

The source code cites the carbon-anode formulation from:

A. Rajkamal and R. Thapa, Advanced Materials Technologies, 2019, 4, 1900307, DOI: 10.1002/admt.201900307.

Chemical-formula mass parser

VQkit contains a built-in atomic-weight table for many common battery-material elements.

The parser supports formulas such as:

LiFePO4
Li2MnSiO4
LiNi0.5Mn0.5O2
C6
C2

The formula must use explicit ordinary element/count notation.

If a composition cannot be parsed safely, enter the molar mass directly.

This design prevents VQkit from silently misinterpreting variable or nonstandard composition notation.

Reference consistency

Battery descriptors are meaningful only when all energy terms use compatible DFT conventions.

For voltage calculations, use consistent:

  • exchange-correlation functional;

  • pseudopotential family;

  • energy cutoff;

  • k-point convergence;

  • spin treatment;

  • Hubbard U where applicable;

  • structural relaxation criteria;

  • reference-cell normalization.

Li metal reference

The implemented voltage expressions use:

E(Li metal) in eV/atom

as the Li/Li+ reference.

For practical calculations, this normally means a properly converged bulk metallic Li calculation using the same computational framework.

Do not mix Li-metal reference energies from incompatible pseudopotentials or functionals.

Formula-unit normalization

The cathode voltage workflow expects energies in:

eV per formula unit

for the compared endpoint compositions.

If supercell energies are used, normalize them consistently before entering the values.

Likewise, the number of inserted Li atoms must correspond to the same chosen formula/cell basis.

0 K approximation

The current battery thermodynamic expressions are mainly electronic-energy calculations.

They do not automatically include:

  • vibrational free energy;

  • configurational entropy;

  • finite-temperature entropy;

  • solvent contributions;

  • electrode/electrolyte interfacial effects;

  • kinetic polarization;

  • concentration-dependent phase coexistence.

Therefore:

\[\Delta G \approx \Delta E\]

is an approximation unless corrections are supplied separately.

Cathode stability versus full phase stability

The lithiation tie-line tool compares one intermediate composition only with a straight line between two chosen endpoint compositions.

It is useful for local relative stability analysis.

It is not equivalent to constructing a full multi-composition convex hull with all competing bulk phases.

For rigorous phase stability, a broader thermodynamic treatment is required.

Diffusion caution

The module contains two different diffusion-related analyses:

Cathode

The cathode calculator can estimate dilute-limit diffusivity using:

\[D = a^2gf x\nu \exp(-E_a/k_BT)\]

Anode

The anode calculator only summarizes supplied NEB migration barriers.

These should not be confused.

The barrier-summary option does not infer diffusivity automatically.

Connection to other VQkit modules

The Batteries module intentionally does not duplicate general analysis.

Use the corresponding VQkit modules for:

NEB

Post-processing
-> Energetics and thermodynamics
-> NEB / transition-state analysis and plot

AIMD diffusion

Post-processing
-> AIMD / trajectory analysis
-> Mean-square displacement
-> Diffusion coefficient

Electronic structure

Post-processing
-> Electronic structure

Structural changes

Post-processing
-> Structural analysis

This separation keeps the battery application module focused on battery-specific derived descriptors.

Current scope

The current battery application module focuses on Li-ion electrode screening.

It does not yet automatically provide specialized workflows for:

  • Na-ion batteries;

  • K-ion batteries;

  • Mg-ion batteries;

  • Li-S reaction networks;

  • solid-state electrolyte stability windows;

  • voltage profiles from full convex-hull enumeration;

  • grand-canonical electrochemical free energies.

Those topics require additional dedicated workflows beyond the currently implemented battery calculator.

Summary

The VQkit Batteries module provides application-oriented analysis for:

Cathodes

  • average Li insertion voltage;

  • theoretical specific capacity;

  • lithiation tie-line stability;

  • volume change;

  • ideal specific energy;

  • dilute-limit Li diffusivity.

Anodes

  • theoretical specific capacity;

  • average Li intercalation voltage / OCV;

  • C6-normalized carbon-anode formation energy;

  • volume expansion;

  • Li migration-barrier statistics.

The calculations are intentionally transparent and formula-based so the user retains control over the DFT reference states and physical assumptions.