Skip to main content

​
Overview

The electric polarizability calculation represents the most significant new finding in the extended capabilities experiments. It demonstrates that the neural backends preserve not only quantum mechanical constraints but also the mathematical structure needed for molecular response properties. From README.md:371:
The polarizability result is the most significant new finding.

​
Calculation Method

​
Hamiltonian with External Field

The total Hamiltonian in the presence of an external electric field F: 
H(F) = H₀ - F·μ
where:
  • H₀: Zero-field molecular Hamiltonian
  • F: External electric field strength (atomic units)
  • μ: Dipole moment operator
From README.md:205-212:
A dipole operator was constructed from PySCF molecular orbitals. The dipole matrix in the MO basis was computed, then mapped to qubits via Jordan-Wigner transformation. The total Hamiltonian at field strength F became: H(F) = H_0 - F * mu where mu is the dipole operator in qubit representation.

​
Dipole Operator Structure

Dipole MO matrix:
[[-3.28091797e-10 -9.27833470e-01]
 [-9.27833470e-01 -3.28091741e-10]]
Structure analysis:
  • Off-diagonal elements: -0.9278
  • Diagonal elements: ~0 (order 10⁻¹⁰)
  • Physical interpretation: Transition dipole between bonding and antibonding orbitals
Jordan-Wigner mapping:
  • 4 Pauli terms
  • Identity contribution: exactly 0.000000
From README.md:39-42:
The dipole operator matrix in the molecular orbital basis showed immediate structure. The off-diagonal elements measured -0.9278, dominating the diagonal elements which were essentially zero. This is the signature of a transition dipole between bonding and antibonding orbitals. The operator contained four Pauli terms with identity contribution exactly zero.

​
Zero-Field Reference

Establishing the zero-field reference is critical for measuring field response. 
============================================================
STEP 1: ZERO-FIELD REFERENCE
============================================================
E(0) = -1.1373060358
θ = [-6.77278357e-09 -1.03333271e-08 -8.66274926e-09 -5.65363690e-08
 -1.11768514e-01]
ΔE_FCI = 2.89e-15
Key metrics:
  • Energy: -1.1373060358 Ha
  • Singles parameters: ~0 (order 10⁻⁸ to 10⁻⁹)
  • Double parameter: -0.111768514
  • Error from FCI: 2.89 × 10⁻¹⁵ Ha
From README.md:46:
This is not approximate. It is exact.

​
Diagnostic Test: Frozen Parameters

To verify that field response requires reoptimization, the zero-field parameters were evaluated at different field strengths: 
--- Diagnostic: same θ, different H(F) ---
  [zero-field θ] F=+0.00000: <H₀>=-1.1373060358, <μ>=0.0000000925, -F<μ>=-0.0000000000, E=-1.1373060358
  [zero-field θ] F=+0.00500: <H₀>=-1.1373060358, <μ>=0.0000000925, -F<μ>=-0.0000000005, E=-1.1373060362
  [zero-field θ] F=+0.01000: <H₀>=-1.1373060358, <μ>=0.0000000925, -F<μ>=-0.0000000009, E=-1.1373060367
  [zero-field θ] F=+0.02000: <H₀>=-1.1373060358, <μ>=0.0000000925, -F<μ>=-0.0000000019, E=-1.1373060376
Observation: With frozen zero-field parameters, the dipole expectation value remains constant (~9.25 × 10⁻⁸) and the energy shift is minimal.From README.md:47:
The dipole expectation value remained essentially constant at 9.25e-08. This tells me the wavefunction was not adapting to the field. It was frozen in its zero-field configuration.

​
Field Sweep with Reoptimization

VQE was run independently at each field value from -0.02 to +0.02 atomic units.

​
Complete Results Table

From RESULTS.md:636-646: 
============================================================
STEP 2: FIELD SWEEP
============================================================
  F=-0.00500: E=-1.1373404123, ΔE=-0.0000343765
  F=+0.00500: E=-1.1373404123, ΔE=-0.0000343765
  F=-0.01000: E=-1.1374435409, ΔE=-0.0001375052
  F=+0.01000: E=-1.1374435409, ΔE=-0.0001375052
  F=-0.01500: E=-1.1376154189, ΔE=-0.0003093832
  F=+0.01500: E=-1.1376154189, ΔE=-0.0003093832
  F=-0.02000: E=-1.1378560417, ΔE=-0.0005500059
  F=+0.02000: E=-1.1378560417, ΔE=-0.0005500059

​
Formatted Data Table

Field F (a.u.)Energy E(F) (Ha)ΔE = E(F) - E(0) (Ha)
-0.02000-1.1378560417-0.0005500059
-0.01500-1.1376154189-0.0003093832
-0.01000-1.1374435409-0.0001375052
-0.00500-1.1373404123-0.0000343765
0.00000-1.13730603580.0000000000
+0.00500-1.1373404123-0.0000343765
+0.01000-1.1374435409-0.0001375052
+0.01500-1.1376154189-0.0003093832
+0.02000-1.1378560417-0.0005500059
Observation: Energy decreases quadratically with field strength, showing the expected polarization response.

​
Symmetry Analysis

The energy response shows perfect inversion symmetry.

​
Symmetry Verification

From RESULTS.md:663-667: 
Symmetry |E(+F)-E(-F)|:
  ±0.0050: 2.22e-16
  ±0.0100: 3.11e-15
  ±0.0150: 2.36e-12
  ±0.0200: 1.11e-15

|E(+0.005) - E(-0.005)|

2.22 × 10⁻¹⁶ Ha

|E(+0.010) - E(-0.010)|

3.11 × 10⁻¹⁵ Ha

|E(+0.015) - E(-0.015)|

2.36 × 10⁻¹² Ha

|E(+0.020) - E(-0.020)|

1.11 × 10⁻¹⁵ Ha
From README.md:49-52:
At F = -0.005, E = -1.1373404123 Ha. At F = +0.005, E = -1.1373404123 Ha. The same value. The symmetry was exact. I checked this carefully. |E(+F) - E(-F)| at F = 0.005 was 2.22e-16. At F = 0.010 it was 3.11e-15. At F = 0.015 it was 2.36e-12. At F = 0.020 it was 1.11e-15. These are machine precision zeros. The energy surface is perfectly symmetric.

​
Physical Interpretation

From README.md:375:
The perfect symmetry |E(+F) - E(-F)| at machine precision tells me that the backends are not introducing spurious field-dependent artifacts. They are not breaking parity. They are not leaking information between positive and negative field directions.

​
Polarizability Extraction

The polarizability is extracted by fitting the energy response to a quadratic form.

​
Fitting Formula

For a molecule in an external field: 
ΔE(F) = -½ α F²
where α is the static electric polarizability. From README.md:215:
Polarizability was extracted by fitting: Delta E(F) = - (1/2) * alpha * F^2

​
Results

From RESULTS.md:669-671: 
  α (VQE)  = 2.7500 a₀³
  α (exact diag, STO-3G) ≈ 2.750 a₀³
  Error = 0.0%

VQE Result

2.7500 a₀³

Exact Diagonalization

2.750 a₀³

Error

0.0%
From README.md:53-58:
Fitting the quadratic form E(F) = E(0) - (1/2) * alpha * F^2 gives alpha = 2.750 a_0^3. I then computed the reference value. Exact diagonalization in the STO-3G basis for H2 gives alpha = 2.750 a_0^3. The error is zero to the precision I can measure.

​
Significance

From README.md:59-62:
Let me explain why this matters. Electric polarizability measures how a molecule responds to an external field. The calculation requires the dipole operator to be correctly constructed from molecular orbitals, the Jordan-Wigner mapping to preserve matrix elements, the VQE optimization to find the correct response wavefunction at each field value, and the energy differences to be computed without systematic bias. Any error in any step propagates to the final answer. The fact that I observe exact agreement with reference calculations tells me the entire pipeline is internally consistent. The neural backends are not just running gates. They are preserving the structure needed for response properties.

​
Pipeline Components

The zero-error result requires correctness at every stage:
  1. PySCF molecular orbital calculation
  2. Dipole operator construction in MO basis
  3. Jordan-Wigner qubit mapping
  4. VQE ansatz construction (UCCSD)
  5. Energy minimization at each field value
  6. Hamiltonian expectation value evaluation
  7. Quadratic fit to extract polarizability
From README.md:378:
The zero-error polarizability tells me that the entire pipeline is internally consistent. The dipole operator construction, the Jordan-Wigner mapping, the VQE optimization, and the energy differencing all work together without introducing bias.

​
Visual Summary

============================================================
POLARIZABILITY ANALYSIS
============================================================

         F               E(F)             ΔE
  -0.02000      -1.1378560417  -0.0005500059
  -0.01500      -1.1376154189  -0.0003093832
  -0.01000      -1.1374435409  -0.0001375052
  -0.00500      -1.1373404123  -0.0000343765
  +0.00000      -1.1373060358  +0.0000000000
  +0.00500      -1.1373404123  -0.0000343765
  +0.01000      -1.1374435409  -0.0001375052
  +0.01500      -1.1376154189  -0.0003093832
  +0.02000      -1.1378560417  -0.0005500059

Symmetry |E(+F)-E(-F)|:
  ±0.0050: 2.22e-16
  ±0.0100: 3.11e-15
  ±0.0150: 2.36e-12
  ±0.0200: 1.11e-15

  α (VQE)  = 2.7500 a₀³
  α (exact diag, STO-3G) ≈ 2.750 a₀³
  Error = 0.0%

​
Limitations

From README.md:395:
Polarizability was tested only for H2 with the STO-3G basis. I do not know if larger molecules or better basis sets preserve this accuracy.

​
Conclusion

From README.md:137-139:
When the fitted polarizability matches exact diagonalization to zero error, this tells me something more. The entire pipeline, from PySCF orbitals to Jordan-Wigner mapping to VQE optimization to energy differencing, is internally consistent. The neural backends slot into this pipeline as drop-in replacements for exact evolution.
From README.md:143:
The backends do not merely preserve probabilities and phases. They preserve the mathematical relationships needed for response calculations.

Build docs developers (and LLMs) love

Get started for freeTalk to us