def test_L1(self):
pcmobj = ddcosmo.DDCOSMO(mol0)
r_vdw = pcmobj.get_atomic_radii()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcmobj.lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol1)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_1 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol2)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_2 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
self.assertAlmostEqual(abs((L_2-L_1)/dx - L1[0,2]).max(), 0, 7)
def test_L1(self):
pcmobj = ddcosmo.DDCOSMO(mol0)
r_vdw = pcmobj.get_atomic_radii()
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ylm_1sph = numpy.vstack(sph.real_sph_vec(coords_1sph, pcmobj.lmax, True))
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol1)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_1 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
pcmobj = ddcosmo.DDCOSMO(mol2)
fi = ddcosmo.make_fi(pcmobj, r_vdw)
L_2 = ddcosmo.make_L(pcmobj, r_vdw, ylm_1sph, fi)
self.assertAlmostEqual(abs((L_2-L_1)/dx - L1[0,2]).max(), 0, 7)
def B1_dot_x(pcmobj, dm, r_vdw, ui, ylm_1sph, cached_pol, L):
mol = pcmobj.mol
mol = pcmobj.mol
natm = mol.natm
nao = mol.nao
lmax = pcmobj.lmax
nlm = (lmax + 1)**2
dms = numpy.asarray(dm)
is_single_dm = dms.ndim == 2
dms = dms.reshape(-1, nao, nao)
n_dm = dms.shape[0]
atom_coords = mol.atom_coords()
atom_charges = mol.atom_charges()
aoslices = mol.aoslice_by_atom()
grids = pcmobj.grids
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = coords_1sph.shape[0]
extern_point_idx = ui > 0
fi0 = ddcosmo.make_fi(pcmobj, r_vdw)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
fi1[:, :, ui == 0] = 0
ui1 = -fi1
Bx = numpy.zeros((natm, 3, nao, nao))
ao = mol.eval_gto('GTOval', grids.coords)
aow = numpy.einsum('gi,g->gi', ao, grids.weights)
aopair = numpy.einsum('gi,gj->gij', ao, aow)
den = numpy.einsum('gij,ij->g', aopair, dm)
psi0 = numpy.zeros((natm, nlm))
i1 = 0
for ia in range(natm):
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
fac_pol = ddcosmo._vstack_factor_fak_pol(fak_pol, lmax)
i0, i1 = i1, i1 + fac_pol.shape[1]
psi0[ia] = -numpy.einsum('mn,n->m', fac_pol, den[i0:i1])
LS0 = numpy.linalg.solve(L.reshape(natm * nlm, -1).T, psi0.ravel())
LS0 = LS0.reshape(natm, nlm)
phi0 = numpy.zeros((natm, nlm))
phi1 = numpy.zeros((natm, 3, natm, nlm))
int3c2e = mol._add_suffix('int3c2e')
int3c2e_ip1 = mol._add_suffix('int3c2e_ip1')
cintopt = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env, int3c2e)
cintopt_ip1 = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env,
int3c2e_ip1)
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
#fakemol = gto.fakemol_for_charges(cav_coords[ui[ia]>0])
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
v_phi = numpy.einsum('pqg,pq->g', v_nj, dm)
phi0[ia] = numpy.einsum('n,ln,n,n->l', weights_1sph, ylm_1sph, ui[ia],
v_phi)
phi1[:, :, ia] += lib.einsum('n,ln,azn,n->azl', weights_1sph, ylm_1sph,
ui1[:, :, ia], v_phi)
Bx += lib.einsum('l,n,ln,azn,ijn->azij', LS0[ia], weights_1sph,
ylm_1sph, ui1[:, :, ia], v_nj)
wtmp = lib.einsum('n,ln,n->ln', weights_1sph, ylm_1sph, ui[ia])
v_e1_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e_ip1,
comp=3,
aosym='s1',
cintopt=cintopt_ip1)
vtmp = lib.einsum('l,ln,xijn->xij', LS0[ia], wtmp, v_e1_nj)
Bx[ia] += vtmp
Bx[ia] += vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
Bx[ja, :, p0:p1, :] -= vtmp[:, p0:p1]
Bx[ja, :, :, p0:p1] -= vtmp[:, p0:p1].transpose(0, 2, 1)
tmp = numpy.einsum('xijn,ij->xn', v_e1_nj[:, p0:p1], dm[p0:p1])
tmp += numpy.einsum('xijn,ji->xn', v_e1_nj[:, p0:p1], dm[:, p0:p1])
phitmp = numpy.einsum('ln,xn->xl', wtmp, tmp)
phi1[ja, :, ia] -= phitmp
phi1[ia, :, ia] += phitmp
Xvec0 = numpy.linalg.solve(L.reshape(natm * nlm, -1), phi0.ravel())
Xvec0 = Xvec0.reshape(natm, nlm)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi0)
phi1 -= lib.einsum('aziljm,jm->azil', L1, Xvec0)
Xvec1 = numpy.linalg.solve(L.reshape(natm * nlm, -1),
phi1.reshape(-1, natm * nlm).T)
Xvec1 = Xvec1.T.reshape(natm, 3, natm, nlm)
psi1 = numpy.zeros((natm, 3, natm, nlm))
i1 = 0
for ia, (coords, weight,
weight1) in enumerate(rks_grad.grids_response_cc(grids)):
i0, i1 = i1, i1 + weight.size
ao = mol.eval_gto('GTOval_sph_deriv1', coords)
aow = numpy.einsum('gi,g->gi', ao[0], weight)
aopair1 = lib.einsum('xgi,gj->xgij', ao[1:], aow)
aow = numpy.einsum('gi,zxg->zxgi', ao[0], weight1)
aopair0 = lib.einsum('zxgi,gj->zxgij', aow, ao[0])
den0 = numpy.einsum('zxgij,ij->zxg', aopair0, dm)
den1 = numpy.empty((natm, 3, weight.size))
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
den1[ja] = numpy.einsum('xgij,ij->xg', aopair1[:, :, p0:p1],
dm[p0:p1, :])
den1[ja] += numpy.einsum('xgij,ji->xg', aopair1[:, :, p0:p1],
dm[:, p0:p1])
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
fac_pol = ddcosmo._vstack_factor_fak_pol(fak_pol, lmax)
scaled_weights = lib.einsum('azm,mn->azn', Xvec1[:, :, ia], fac_pol)
scaled_weights *= weight
aow = numpy.einsum('gi,azg->azgi', ao[0], scaled_weights)
Bx -= numpy.einsum('gi,azgj->azij', ao[0], aow)
tmp = numpy.einsum('mn,zxn->zxm', fac_pol, den1)
psi1[:, :, ia] -= numpy.einsum('mn,zxn->zxm', fac_pol, den0)
psi1[ia, :, ia] -= tmp.sum(axis=0)
for ja in range(natm):
psi1[ja, :, ia] += tmp[ja]
eta_nj = lib.einsum('mn,m->n', fac_pol, Xvec0[ia])
Bx -= lib.einsum('n,zxnpq->zxpq', eta_nj, aopair0)
vtmp = lib.einsum('n,xnpq->xpq', eta_nj, aopair1)
Bx[ia] -= vtmp
Bx[ia] -= vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, q0, q1 = aoslices[ja]
Bx[ja, :, q0:q1, :] += vtmp[:, q0:q1]
Bx[ja, :, :, q0:q1] += vtmp[:, q0:q1].transpose(0, 2, 1)
psi1 -= numpy.einsum('il,aziljm->azjm', LS0, L1)
LS1 = numpy.linalg.solve(
L.reshape(natm * nlm, -1).T,
psi1.reshape(-1, natm * nlm).T)
LS1 = LS1.T.reshape(natm, 3, natm, nlm)
cav_coords = (atom_coords.reshape(natm, 1, 3) +
numpy.einsum('r,gx->rgx', r_vdw, coords_1sph))
cav_coords = cav_coords[extern_point_idx]
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
v_phi = numpy.zeros((natm, ngrid_1sph, nao, nao))
v_phi[extern_point_idx] += v_nj.transpose(2, 0, 1)
Bx += lib.einsum('azjx,n,xn,jn,jnpq->azpq', LS1, weights_1sph, ylm_1sph,
ui, v_phi)
return Bx
def make_B1(pcmobj, r_vdw, ui, ylm_1sph, cached_pol, L):
'''1st order'''
mol = pcmobj.mol
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = coords_1sph.shape[0]
mol = pcmobj.mol
natm = mol.natm
nao = mol.nao
lmax = pcmobj.lmax
nlm = (lmax + 1)**2
atom_coords = mol.atom_coords()
atom_charges = mol.atom_charges()
grids = pcmobj.grids
extern_point_idx = ui > 0
cav_coords = (atom_coords.reshape(natm, 1, 3) +
numpy.einsum('r,gx->rgx', r_vdw, coords_1sph))
cav_coords = cav_coords[extern_point_idx]
int3c2e = mol._add_suffix('int3c2e')
cintopt = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env, int3c2e)
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
nao_pair = v_nj.shape[0]
v_phi = numpy.zeros((natm, ngrid_1sph, nao, nao))
v_phi[extern_point_idx] += v_nj.transpose(2, 0, 1)
phi0 = numpy.einsum('n,xn,jn,jnpq->jxpq', weights_1sph, ylm_1sph, ui,
v_phi)
Xvec0 = numpy.linalg.solve(L.reshape(natm * nlm, -1),
phi0.reshape(natm * nlm, -1))
Xvec0 = Xvec0.reshape(natm, nlm, nao, nao)
ao = mol.eval_gto('GTOval', grids.coords)
aow = numpy.einsum('gi,g->gi', ao, grids.weights)
aopair = numpy.einsum('gi,gj->gij', ao, aow)
psi0 = numpy.zeros((natm, nlm, nao, nao))
i1 = 0
for ia in range(natm):
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
i0, i1 = i1, i1 + fak_pol[0].shape[1]
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
psi0[ia, p0:p1] = -fac * numpy.einsum('mn,nij->mij', fak_pol[l],
aopair[i0:i1])
fi0 = ddcosmo.make_fi(pcmobj, r_vdw)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
fi1[:, :, ui == 0] = 0
ui1 = -fi1
phi1 = numpy.zeros(ui1.shape[:3] + (nlm, nao, nao))
int3c2e = mol._add_suffix('int3c2e')
int3c2e_ip1 = mol._add_suffix('int3c2e_ip1')
aoslices = mol.aoslice_by_atom()
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
#fakemol = gto.fakemol_for_charges(cav_coords[ui[ia]>0])
fakemol = gto.fakemol_for_charges(cav_coords)
v_nj = df.incore.aux_e2(mol, fakemol, intor=int3c2e, aosym='s1')
phi1[:, :, ia] += lib.einsum('n,ln,azn,ijn->azlij', weights_1sph,
ylm_1sph, ui1[:, :, ia], v_nj)
v_e1_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e_ip1,
comp=3,
aosym='s1')
v_e2_nj = v_e1_nj + v_e1_nj.transpose(0, 2, 1, 3)
phi1[ia, :, ia] += lib.einsum('n,ln,n,xijn->xlij', weights_1sph,
ylm_1sph, ui[ia], v_e2_nj)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
v = numpy.einsum('n,ln,n,xijn->xlij', weights_1sph, ylm_1sph,
ui[ia], v_e1_nj[:, p0:p1])
phi1[ja, :, ia, :, p0:p1, :] -= v
phi1[ja, :, ia, :, :, p0:p1] -= v.transpose(0, 1, 3, 2)
psi1 = numpy.zeros((natm, 3, natm, nlm, nao, nao))
i1 = 0
for ia, (coords, weight,
weight1) in enumerate(rks_grad.grids_response_cc(grids)):
i0, i1 = i1, i1 + weight.size
ao = mol.eval_gto('GTOval_sph_deriv1', coords)
aow = numpy.einsum('gi,g->gi', ao[0], weight)
aopair1 = lib.einsum('xgi,gj->xgij', ao[1:], aow)
aow = numpy.einsum('gi,zxg->zxgi', ao[0], weight1)
aopair0 = lib.einsum('zxgi,gj->zxgij', aow, ao[0])
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
psi1[:, :, ia, p0:p1] -= fac * numpy.einsum(
'mn,zxnij->zxmij', fak_pol[l], aopair0)
vtmp = fac * numpy.einsum('mn,xnij->xmij', fak_pol[l], aopair1)
psi1[ia, :, ia, p0:p1] -= vtmp
psi1[ia, :, ia, p0:p1] -= vtmp.transpose(0, 1, 3, 2)
for ja in range(natm):
shl0, shl1, q0, q1 = aoslices[ja]
psi1[ja, :, ia, p0:p1, q0:q1, :] += vtmp[:, :, q0:q1]
psi1[ja, :, ia, p0:p1, :,
q0:q1] += vtmp[:, :, q0:q1].transpose(0, 1, 3, 2)
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi0)
Xvec1 = numpy.linalg.solve(
L.reshape(natm * nlm, -1),
phi1.transpose(2, 3, 0, 1, 4, 5).reshape(natm * nlm, -1))
Xvec1 = Xvec1.reshape(natm, nlm, natm, 3, nao,
nao).transpose(2, 3, 0, 1, 4, 5)
LS0 = numpy.linalg.solve(
L.reshape(natm * nlm, -1).T, psi0.reshape(natm * nlm, -1))
LS0 = LS0.reshape(natm, nlm, nao, nao)
B = lib.einsum('ixnlpq,nlrs->ixpqrs', psi1, Xvec0)
B += lib.einsum('nlpq,ixnlrs->ixpqrs', psi0, Xvec1)
B -= lib.einsum('ilpq,aziljm,jmrs->azpqrs', LS0, L1, Xvec0)
B = B + B.transpose(0, 1, 4, 5, 2, 3)
return B
def nuc_part1(pcmobj, r_vdw, ui, ylm_1sph, cached_pol, L):
'''1st order'''
mol = pcmobj.mol
natm = mol.natm
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = coords_1sph.shape[0]
lmax = pcmobj.lmax
nlm = (lmax + 1)**2
nao = mol.nao
atom_coords = mol.atom_coords()
atom_charges = mol.atom_charges()
grids = pcmobj.grids
aoslices = mol.aoslice_by_atom()
extern_point_idx = ui > 0
fi0 = ddcosmo.make_fi(pcmobj, r_vdw)
fi1 = ddcosmo_grad.make_fi1(pcmobj, pcmobj.get_atomic_radii())
fi1[:, :, ui == 0] = 0
ui1 = -fi1
vmat1 = numpy.zeros((natm, 3, nao, nao))
cav_coords = (atom_coords.reshape(natm, 1, 3) +
numpy.einsum('r,gx->rgx', r_vdw, coords_1sph))
v_phi = numpy.zeros((natm, ngrid_1sph))
for ia in range(natm):
# Note (-) sign is not applied to atom_charges, because (-) is explicitly
# included in rhs and L matrix
d_rs = atom_coords.reshape(-1, 1, 3) - cav_coords[ia]
v_phi[ia] = numpy.einsum('z,zp->p', atom_charges,
1. / lib.norm(d_rs, axis=2))
phi0 = -numpy.einsum('n,xn,jn,jn->jx', weights_1sph, ylm_1sph, ui, v_phi)
Xvec0 = numpy.linalg.solve(L.reshape(natm * nlm, -1), phi0.ravel())
Xvec0 = Xvec0.reshape(natm, nlm)
ngrid_1sph = weights_1sph.size
v_phi0 = numpy.empty((natm, ngrid_1sph))
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
d_rs = atom_coords.reshape(-1, 1, 3) - cav_coords
v_phi0[ia] = numpy.einsum('z,zp->p', atom_charges,
1. / lib.norm(d_rs, axis=2))
phi1 = -numpy.einsum('n,ln,azjn,jn->azjl', weights_1sph, ylm_1sph, ui1,
v_phi0)
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
for ja in range(natm):
rs = atom_coords[ja] - cav_coords
d_rs = lib.norm(rs, axis=1)
v_phi = atom_charges[ja] * numpy.einsum('px,p->px', rs,
1. / d_rs**3)
tmp = numpy.einsum('n,ln,n,nx->xl', weights_1sph, ylm_1sph, ui[ia],
v_phi)
phi1[ja, :, ia] += tmp # response of the other atoms
phi1[ia, :, ia] -= tmp # response of cavity grids
L1 = ddcosmo_grad.make_L1(pcmobj, r_vdw, ylm_1sph, fi0)
phi1 -= lib.einsum('aziljm,jm->azil', L1, Xvec0)
Xvec1 = numpy.linalg.solve(L.reshape(natm * nlm, -1),
phi1.reshape(-1, natm * nlm).T)
Xvec1 = Xvec1.T.reshape(natm, 3, natm, nlm)
i1 = 0
for ia, (coords, weight,
weight1) in enumerate(rks_grad.grids_response_cc(grids)):
i0, i1 = i1, i1 + weight.size
ao = mol.eval_gto('GTOval_sph_deriv1', coords)
aow = numpy.einsum('gi,g->gi', ao[0], weight)
aopair1 = lib.einsum('xgi,gj->xgij', ao[1:], aow)
aow = numpy.einsum('gi,zxg->zxgi', ao[0], weight1)
aopair0 = lib.einsum('zxgi,gj->zxgij', aow, ao[0])
fak_pol, leak_idx = cached_pol[mol.atom_symbol(ia)]
fac_pol = ddcosmo._vstack_factor_fak_pol(fak_pol, lmax)
vmat1 -= numpy.einsum('m,mn,zxnij->zxij', Xvec0[ia], fac_pol, aopair0)
vtmp = numpy.einsum('m,mn,xnij->xij', Xvec0[ia], fac_pol, aopair1)
vmat1[ia, :] -= vtmp
vmat1[ia, :] -= vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
vmat1[ja, :, p0:p1, :] += vtmp[:, p0:p1]
vmat1[ja, :, :, p0:p1] += vtmp[:, p0:p1].transpose(0, 2, 1)
scaled_weights = lib.einsum('azm,mn->azn', Xvec1[:, :, ia], fac_pol)
scaled_weights *= weight
aow = numpy.einsum('gi,azg->azgi', ao[0], scaled_weights)
vmat1 -= numpy.einsum('gi,azgj->azij', ao[0], aow)
psi0 = numpy.zeros((natm, nlm))
for ia in range(natm):
psi0[ia,
0] += numpy.sqrt(4 * numpy.pi) / r_vdw[ia] * mol.atom_charge(ia)
LS0 = numpy.linalg.solve(L.reshape(natm * nlm, -1).T, psi0.ravel())
LS0 = LS0.reshape(natm, nlm)
LS1 = numpy.einsum('il,aziljm->azjm', LS0, L1)
LS1 = numpy.linalg.solve(
L.reshape(natm * nlm, -1).T,
LS1.reshape(-1, natm * nlm).T)
LS1 = LS1.T.reshape(natm, 3, natm, nlm)
int3c2e = mol._add_suffix('int3c2e')
int3c2e_ip1 = mol._add_suffix('int3c2e_ip1')
cintopt = gto.moleintor.make_cintopt(mol._atm, mol._bas, mol._env, int3c2e)
for ia in range(natm):
cav_coords = atom_coords[ia] + r_vdw[ia] * coords_1sph
#fakemol = gto.fakemol_for_charges(cav_coords[ui[ia]>0])
fakemol = gto.fakemol_for_charges(cav_coords)
wtmp = lib.einsum('l,n,ln->ln', LS0[ia], weights_1sph, ylm_1sph)
v_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e,
aosym='s1',
cintopt=cintopt)
vmat1 -= numpy.einsum('azl,n,ln,n,pqn->azpq', LS1[:, :, ia],
weights_1sph, ylm_1sph, ui[ia], v_nj)
vmat1 += lib.einsum('ln,azn,ijn->azij', wtmp, ui1[:, :, ia], v_nj)
v_e1_nj = df.incore.aux_e2(mol,
fakemol,
intor=int3c2e_ip1,
comp=3,
aosym='s1')
vtmp = lib.einsum('ln,n,xijn->xij', wtmp, ui[ia], v_e1_nj)
vmat1[ia] += vtmp
vmat1[ia] += vtmp.transpose(0, 2, 1)
for ja in range(natm):
shl0, shl1, p0, p1 = aoslices[ja]
vmat1[ja, :, p0:p1, :] -= vtmp[:, p0:p1]
vmat1[ja, :, :, p0:p1] -= vtmp[:, p0:p1].transpose(0, 2, 1)
return vmat1